Experimental Probability

Probability can tell us what we expect to happen, but real experiments do not always produce exactly the expected results.

A fair coin has a theoretical probability of:

P(heads) = 1/2 = 0.5 = 50%

Does that mean that if we flip the coin 10 times, we will always get exactly 5 heads?

No.

We might get:

• 4 heads and 6 tails
• 7 heads and 3 tails
• 2 heads and 8 tails
• or even 10 heads and no tails

Experimental probability uses the results of an actual experiment to estimate how likely an event is.

Instead of asking:

"What should happen mathematically?"

we ask:

"What actually happened when we tried it?"

What Is Experimental Probability?

Experimental probability is probability calculated from the results of an experiment or observation.

It is also sometimes called empirical probability.

The basic equation is:

Experimental probability = number of times the event occurs / total number of trials

For example, suppose a coin is flipped 20 times.

Heads occurs 12 times.

Experimental probability of heads:

P(heads) = 12/20

= 3/5

= 0.6

= 60%

This does not mean the theoretical probability of heads has changed.

For a fair coin, the theoretical probability is still:

50%

The experimental result simply happened to be 60% in this particular experiment.

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What Is a Probability Experiment?

A probability experiment is a repeatable process that can produce different outcomes.

Examples include:

• flipping a coin
• rolling a die
• spinning a spinner
• drawing coloured counters from a bag
• selecting cards
• dropping objects and observing how they land
• observing whether seeds germinate
• recording whether buses arrive on time

Each time the experiment is performed is called a trial.

If a coin is flipped 50 times:

Number of trials = 50

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Outcomes and Events

An outcome is one possible result of a trial.

For a coin:

Heads or Tails

For a six-sided die:

1, 2, 3, 4, 5, or 6

An event is the result or group of results we are interested in.

For example:

Event: rolling an even number

Successful outcomes:

2, 4, 6

Experimental probability measures how frequently the event actually occurs during repeated trials.

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Conducting a Probability Experiment

A good probability experiment should follow a clear procedure.

Decide what you are investigating

For example:

What is the experimental probability of rolling a 6?

Choose the number of trials

For example:

60 rolls

Conduct the experiment consistently

Roll the same die in approximately the same way each time.

Record every result

Do not record only the outcomes you want.

Organize the data

Use a tally chart or frequency table.

Calculate experimental probability

Use:

Experimental probability = event frequency / total trials

Compare with theoretical probability

Ask:

How close are the experimental and theoretical results?

Interpret the differences

Consider random variation and the number of trials.

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Recording Experimental Results

Suppose a die is rolled 30 times.

A useful table might look like this:

Check the total:

4 + 5 + 6 + 3 + 5 + 7 = 30

Recording results carefully is important because every later calculation depends on the quality of the original data.

Frequency

The frequency of an outcome is the number of times it occurs.

Suppose a spinner is spun 40 times.

Results:

Red = 18
Blue = 12
Green = 10

The frequency of red is:

18

The total frequency should equal the number of trials:

18 + 12 + 10 = 40

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Relative Frequency

Relative frequency compares the frequency of an outcome with the total number of trials.

Relative frequency = frequency / total number of trials

For red:

Relative frequency = 18/40

= 0.45

= 45%

Relative frequency is essentially the experimental probability of that outcome.

Therefore:

Experimental probability and relative frequency are closely related ideas.

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Worked Example: Coin Flips

A student flips a coin 50 times.

Results:

Heads = 28

Tails = 22

Experimental probability of heads

P(heads) = 28/50

= 0.56

= 56%

Experimental probability of tails

P(tails) = 22/50

= 0.44

= 44%

Check:

0.56 + 0.44 = 1

or:

56% + 44% = 100%

The experimental probabilities account for all possible outcomes.

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Comparing with Theoretical Probability

For a fair coin:

Theoretical P(heads) = 0.5

Experimental P(heads) = 0.56

Difference:

0.56 − 0.50 = 0.06

or:

6 percentage points

The experimental probability is slightly higher than the theoretical probability.

This does not necessarily indicate anything unusual.

Random variation is expected.

Theoretical Probability

Theoretical probability is based on mathematical reasoning about possible outcomes.

For a fair six-sided die:

P(rolling 1) = 1/6

P(rolling 2) = 1/6

P(rolling 3) = 1/6

and so on.

As a percentage:

1/6 ≈ 16.7%

We do not need to roll the die to calculate this.

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Experimental Probability

Suppose we actually roll the die 60 times.

The number 6 appears:

13 times

Experimental probability:

P(6) = 13/60

≈ 0.217

≈ 21.7%

Theoretical probability:

P(6) = 1/6

≈ 16.7%

The values differ.

That is completely possible.

Experimental probability is based on observed results, while theoretical probability is based on the expected mathematical model.

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Expected Outcomes

Theoretical probability can help predict how many times we might expect an event to occur.

Suppose:

P(rolling a 6) = 1/6

and we roll a die:

60 times

Expected number of sixes:

60 × 1/6 = 10

We would expect approximately:

10 sixes

But this does not mean we must get exactly 10.

We might obtain:

8, 9, 10, 11, 13, or another value.

The expected value describes what we would anticipate based on probability, not a guaranteed result.

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Worked Example: Expected Heads

A fair coin is flipped:

100 times

Theoretical probability of heads:

P(heads) = 0.5

Expected heads:

100 × 0.5 = 50

Suppose the experiment produces:

54 heads

Experimental probability:

54/100 = 0.54

= 54%

The experimental result differs from the theoretical probability by:

54% − 50% = 4 percentage points

This is an example of ordinary experimental variation.

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Why Don't Experimental Results Match Exactly?

Random processes naturally produce variation.

Suppose two students each flip a fair coin 20 times.

Student A

Heads = 8

P(heads) = 8/20 = 0.40

Student B

Heads = 13

P(heads) = 13/20 = 0.65

Both students used the same type of experiment.

Their results differ because random outcomes do not have to follow the theoretical probability exactly over a small number of trials.

This is called random variation.

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Random Does Not Mean "Anything Goes"

Random results are unpredictable individually, but they often show patterns over many trials.

For example, a fair coin might produce:

HHHHTHHH

over eight flips.

That may look heavily biased toward heads.

But over:

• 100 flips
• 1,000 flips
• 10,000 flips

the proportion of heads often moves closer to:

0.5

Randomness can produce short-term patterns without changing the underlying probability.

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Increasing the Number of Trials

Consider a fair coin.

After 10 flips

Heads = 7

Experimental probability:

7/10 = 0.70

After 100 flips

Heads = 54

Experimental probability:

54/100 = 0.54

After 1,000 flips

Heads = 507

Experimental probability:

507/1000 = 0.507

Notice what happens:

0.70 → 0.54 → 0.507

The experimental probability becomes closer to:

0.5

as the number of trials increases.

This is an important principle of probability.

The Law of Large Numbers

The law of large numbers states that as the number of independent trials becomes very large, experimental probability tends to become closer to the theoretical probability.

For example:

Theoretical probability of heads = 0.5

After a small number of flips, experimental results may be far from 0.5.

After many flips, the proportion will often become closer to 0.5.

This does not mean it must become exactly 0.500000.

Random variation still exists.

The key idea is:

More trials generally produce a more stable estimate of probability.

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Small Samples Can Be Misleading

Suppose you flip a coin twice.

You get:

Heads, Heads

Experimental probability of heads:

2/2 = 1

= 100%

Should you conclude:

"This coin always lands heads"?

No.

Two trials provide very little evidence.

Now suppose you flip the coin:

10,000 times

and obtain:

9,950 heads

That would provide much stronger evidence that the coin may not behave like a fair coin.

The number of trials matters when interpreting experimental probability.

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Experimental Probability with a Spinner

Suppose a spinner has four equal sections:

Red, Blue, Green, Yellow

The theoretical probability of each colour is:

1/4 = 0.25 = 25%

A student spins it 40 times:

Experimental probabilities:

P(red) = 8/40 = 0.20

P(blue) = 12/40 = 0.30

P(green) = 9/40 = 0.225

P(yellow) = 11/40 = 0.275

None is exactly:

0.25

That does not automatically mean the spinner is unfair.

Forty trials still allow considerable random variation.

Experimental Probability with Dice

Suppose a die is rolled 120 times.

Results:

Total:

19 + 18 + 21 + 22 + 17 + 23 = 120

For a fair die, the expected frequency of each result is:

120 × 1/6 = 20

The actual frequencies range from:

17 to 23

This variation is not surprising.

Random experiments rarely produce perfectly equal frequencies.

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Expected Frequency

The expected frequency tells us approximately how often an event should occur based on theoretical probability.

Expected frequency = probability × number of trials

For example:

A fair die is rolled 300 times.

P(rolling 4) = 1/6

Expected frequency:

300 × 1/6 = 50

We would expect approximately:

50 fours

But the actual number might be:

47, 52, 55, or another nearby value.

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Worked Example: Coloured Counters

A bag contains:

4 red counters
3 blue counters
3 green counters

Total = 10

Theoretical probabilities:

P(red) = 4/10 = 0.4

P(blue) = 3/10 = 0.3

P(green) = 3/10 = 0.3

Suppose a counter is randomly selected, its colour recorded, and then replaced before the next trial.

After 100 trials:

Red = 43
Blue = 28
Green = 29

Experimental probabilities:

P(red) = 43/100 = 0.43

P(blue) = 28/100 = 0.28

P(green) = 29/100 = 0.29

These values are reasonably close to the theoretical probabilities.

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Why Replacement Matters

Suppose a counter is drawn from a bag.

If you replace it before the next draw, the contents of the bag remain the same.

Therefore, the probabilities remain unchanged from trial to trial.

If you do not replace it, the contents change.

That means the probability for the next trial also changes.

For a simple repeated probability experiment, replacement is often used to keep the conditions consistent.

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Fair Experiments

Probability experiments should be designed carefully.

For example, when investigating a spinner:

• spin it properly each time
• avoid deliberately controlling where it stops
• use the same spinner
• record every result
• conduct enough trials

For dice:

• use the same die
• roll rather than deliberately place it
• use an appropriate surface
• record every outcome

For counters:

• mix them properly
• select without looking
• replace them if the experiment requires constant probabilities

Good experimental design helps reduce bias.

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Bias in Probability Experiments

Bias occurs when the experimental method systematically favours certain outcomes.

For example:

• a spinner is uneven
• a die is weighted
• counters are not mixed properly
• the experimenter can see inside the bag
• some results are not recorded
• unsuccessful trials are deliberately ignored

Bias is different from random variation.

Random variation

Produces unpredictable differences that fluctuate.

Bias

Systematically pushes results in a particular direction.

This distinction is important when evaluating experimental results.

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Does a Difference Prove Something Is Unfair?

Suppose a coin is flipped 20 times and produces:

14 heads

Experimental probability:

14/20 = 0.70

The theoretical probability for a fair coin is:

0.50

Does this prove the coin is unfair?

No.

Twenty trials may produce substantial random variation.

A sensible next step would be to:

conduct more trials

If thousands of flips continued to produce results far from 50%, then there would be stronger evidence that the coin or experimental method should be investigated.

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Comparing Two Experiments

Suppose two groups investigate a coin.

Group A

20 flips

Heads = 14

Experimental probability:

14/20 = 0.70

Group B

1,000 flips

Heads = 515

Experimental probability:

515/1000 = 0.515

The theoretical probability is:

0.5

Group B's estimate is much closer.

Its larger number of trials provides a more stable estimate of the coin's probability.

This demonstrates why scientists and statisticians value larger samples.

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Organizing Data with a Frequency Table

Suppose a die is rolled 50 times.

Rather than recording a long list such as:

3, 1, 6, 2, 2, 5, 4, 3...

we can summarize the results:

Check:

Experimental probabilities should total:

1

0.14 + 0.20 + 0.16 + 0.18 + 0.16 + 0.16 = 1.00

This is a useful way to check calculations.

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Graphing Experimental Results

Graphs can make experimental probability easier to interpret.

Useful displays include:

• bar graphs
• frequency charts
• line graphs showing probability over increasing numbers of trials

For example, a bar graph could compare:

theoretical frequency

with:

experimental frequency

for each die result.

A line graph could show how:

P(heads)

changes after:

10, 20, 50, 100, 500, and 1,000 flips.

This can make the effect of increasing trial numbers very clear.

Simulation

Probability experiments do not always need physical objects.

We can use computer simulations to generate random outcomes.

A computer can simulate:

• coin flips
• dice rolls
• card draws
• spinner results
• random selections

This is particularly useful when we want to conduct:

thousands or millions of trials

very quickly.

For example, a computer could simulate 100,000 coin flips and calculate the experimental probability of heads.

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Physical Experiment vs Simulation

Suppose we want to investigate rolling a die.

Physical experiment

Actually roll a die repeatedly.

Advantages:

• hands-on
• demonstrates real experimental variation
• may reveal imperfections in a real die

Computer simulation

Generate random integers from 1 to 6.

Advantages:

• extremely fast
• easy to conduct many trials
• easy to repeat

Both can be useful.

The appropriate method depends on the question being investigated.

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Experimental Probability in Real Life

Experimental probability is not limited to classroom games.

It can be estimated from real-world data.

For example:

Transport

Out of 200 recorded bus journeys, 170 arrived within five minutes of schedule.

Experimental probability:

170/200 = 0.85

= 85%

Manufacturing

Out of 5,000 products, 35 were defective.

Experimental probability of a defect:

35/5000

= 0.007

= 0.7%

Sports

A player makes 76 of 100 free throws.

Experimental probability of success:

76/100 = 0.76

= 76%

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Experimental Probability in Science

Scientists frequently estimate probabilities from observed data.

Examples include:

• proportion of seeds that germinate
• proportion of offspring showing a trait
• probability of equipment failure
• frequency of a particular weather event
• survival rates in ecological studies
• proportion of samples showing a particular characteristic

These probabilities are based on observations.

More data generally provide a stronger estimate of the underlying probability.

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Worked Example: Seed Germination

A scientist plants:

200 seeds

After the observation period:

174 germinate

Experimental probability of germination:

P(germination) = 174/200

= 0.87

= 87%

Experimental probability of not germinating:

26/200

= 0.13

= 13%

Check:

87% + 13% = 100%

Making Predictions from Experimental Probability

Experimental probability can be used to make predictions.

Suppose a basketball player makes:

42 of 60 free throws

Experimental probability of success:

42/60 = 0.7

If the player attempts another 100 similar free throws, we might predict:

100 × 0.7 = 70 successful shots

This is an estimate.

It does not guarantee exactly 70 successes.

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Worked Example: Predicting Future Outcomes

A factory inspects 500 components.

20 are defective.

Experimental probability of a defect:

20/500 = 0.04

= 4%

If similar conditions continue and the factory produces another 2,000 components:

Expected defective components:

2000 × 0.04 = 80

We might predict approximately:

80 defective components

But the actual number could be higher or lower.

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Comparing Experimental Results

Suppose three groups roll the same type of fair die.

Group A

12 trials

P(6) = 0.25

Group B

120 trials

P(6) = 0.183

Group C

12,000 trials

P(6) = 0.168

The theoretical probability is:

1/6 ≈ 0.167

The largest experiment gives an experimental probability very close to the theoretical value.

This illustrates a general pattern:

Larger numbers of trials tend to produce more stable experimental probabilities.

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When Results Are Unexpected

Suppose a six-sided die is rolled:

10,000 times

and the number 6 appears:

3,500 times

Experimental probability:

3500/10000 = 0.35

The theoretical probability for a fair die is approximately:

0.167

That is a substantial difference after many trials.

Possible explanations should be investigated:

• Is the die fair?
• Was the method biased?
• Was the data recorded correctly?
• Was the simulation programmed correctly?
• Were all outcomes included?

Unexpected results can provide a reason for further investigation.

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Experimental Variation Is Useful Information

Variation between experiments is not simply an inconvenience.

It teaches us something important about probability.

Suppose five groups each flip a coin 50 times.

They might obtain:

Group A: 44% heads
Group B: 54% heads
Group C: 48% heads
Group D: 58% heads
Group E: 52% heads

Different groups obtain different results even though the theoretical probability is the same.

This demonstrates that:

experimental probability itself varies from sample to sample.

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Combining Experimental Results

Suppose five groups each conduct 100 coin flips.

Their numbers of heads are:

48, 53, 45, 51, 55

Total heads:

48 + 53 + 45 + 51 + 55 = 252

Total flips:

5 × 100 = 500

Combined experimental probability:

252/500

= 0.504

= 50.4%

Combining results increases the number of trials and can produce a more stable estimate.

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Common Mistakes

Using the wrong denominator

Experimental probability uses:

number of times event occurred / total number of trials

not the number of possible theoretical outcomes.

Confusing theoretical and experimental probability

Theoretical probability comes from a mathematical model.

Experimental probability comes from actual observations.

Expecting experimental probability to match exactly

Random variation is normal.

Assuming 50% means exactly half every time

It describes long-term likelihood, not a guaranteed short-term result.

Drawing conclusions from too few trials

Small samples can produce highly variable results.

Changing or ignoring results

Every valid trial should be recorded honestly.

Assuming a difference proves bias

Random variation can create differences, particularly in small samples.

Forgetting replacement

Removing an object without replacing it can change probabilities between trials.

Forgetting to total the frequencies

Frequencies should add to the total number of trials.

Assuming more trials eliminate randomness

Randomness remains, but relative frequencies tend to become more stable.

Saying expected frequency is guaranteed

Expected values are predictions, not promises.

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Key Terms

Experimental probability — Probability calculated from observed experimental results.

Empirical probability — Another name for probability estimated from observations.

Theoretical probability — Probability calculated mathematically from a model of possible outcomes.

Probability experiment — A repeatable process that can produce uncertain outcomes.

Trial — One performance of a probability experiment.

Outcome — One possible result of a trial.

Event — An outcome or collection of outcomes being investigated.

Frequency — The number of times an outcome occurs.

Relative frequency — The frequency of an outcome divided by the total number of trials.

Expected outcome — A result anticipated from a probability model.

Expected frequency — The predicted number of times an event should occur over a given number of trials.

Random — Describing an outcome that cannot be predicted with certainty before a trial.

Random variation — Natural differences between results produced by random processes.

Sample size — The number of observations or trials used.

Law of large numbers — The principle that experimental relative frequencies tend to approach theoretical probabilities as the number of trials becomes very large.

Fair experiment — A probability experiment designed so that outcomes are not unintentionally favoured.

Bias — A systematic influence that favours certain outcomes.

Replacement — Returning a selected object before the next trial so that probabilities remain unchanged.

Simulation — A model, often computer-based, used to imitate a probability experiment.

Data table — An organized display of experimental observations.

Tally — A method of recording occurrences as they happen.

Frequency table — A table showing how often each outcome occurs.

Long-run probability — The proportion an event tends toward over a very large number of trials.

Prediction — An estimate of a future result based on available evidence or probability.

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Key Takeaways

• Experimental probability is based on what actually happens during trials.
• Calculate it using:

Experimental probability = event frequency / total number of trials

• Experimental probability is also called empirical probability.
• A trial is one repetition of a probability experiment.
• Results should be recorded carefully and completely.
• Tally charts and frequency tables help organize experimental results.
• Relative frequency and experimental probability represent essentially the same calculation.
• Theoretical probability describes what a mathematical model predicts.
• Experimental probability describes what was observed.
• Experimental and theoretical probabilities do not need to match exactly.
• Random variation causes experimental results to differ between experiments.
• Small numbers of trials often produce highly variable experimental probabilities.
• Increasing the number of trials generally makes experimental probability more stable.
• Over many trials, experimental probability often approaches theoretical probability.
• This pattern is described by the law of large numbers.
• Expected frequency can be calculated using:

Expected frequency = probability × number of trials

• Expected frequency is not a guaranteed result.
• A difference between experimental and theoretical probability does not automatically prove that an object is unfair.
• Large differences that persist across many trials may deserve investigation.
• Bias is different from random variation.
• A fair experimental method should avoid systematically favouring outcomes.
• Replacement may be necessary to keep probabilities constant between trials.
• Simulations allow very large probability experiments to be conducted quickly.
• Experimental probabilities can be used to predict future outcomes.
• Predictions based on experimental probability are estimates, not guarantees.
• Combining data from several experiments can provide a larger sample and more stable estimate.
• Experimental probability is used in science, manufacturing, sports, transport, biology, and many other real-world settings.

When conducting a probability experiment, think:

Question → Trials → Record → Organize → Calculate → Compare → Explain

And always ask:

How many trials were conducted?

Were the results recorded fairly?

What is the experimental probability?

What is the theoretical probability or expected outcome?

How different are they?

Could the difference simply be random variation?

Would more trials give us stronger evidence?

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Check Your Understanding

1. Define experimental probability.

2. Write the equation used to calculate experimental probability.

3. A coin is flipped 40 times and lands heads 23 times. Calculate the experimental probability of heads as:

a. a fraction
b. a decimal
c. a percentage

4. Calculate the experimental probability of tails in Question 3.

5. Compare the experimental probability of heads in Question 3 with the theoretical probability.

6. Does the difference prove that the coin is unfair? Explain.

7. A die is rolled 60 times. The number 6 occurs 13 times. Calculate the experimental probability of rolling a 6.

8. What is the theoretical probability of rolling a 6?

9. Why might your answers to Questions 7 and 8 differ?

10. A spinner is spun 100 times.

Results:

Calculate the experimental probability of each colour.

11. If the spinner in Question 10 has four equal sections, what is the theoretical probability of each colour?

12. Explain why the experimental results do not have to be exactly 25 times for each colour.

13. A fair coin is flipped 200 times. How many heads would you expect?

14. If 108 heads actually occur, calculate the experimental probability.

15. Explain the difference between expected frequency and actual frequency.

16. Why do larger numbers of trials generally produce more stable estimates of probability?

17. A bag contains 3 red and 2 blue counters. A counter is selected and replaced after every trial. Calculate the theoretical probability of selecting red.

18. The experiment in Question 17 is conducted 100 times and red is selected 57 times. Calculate the experimental probability and compare it with the theoretical probability.

19. Explain why replacement is important if you want the probability to remain the same during every trial.

20. Two students investigate the same coin.

Student A flips it 10 times and gets 8 heads.

Student B flips it 1,000 times and gets 508 heads.

Calculate the experimental probability for each student. Which result provides the more stable estimate of the coin's probability, and why?

21. Explain the difference between random variation and bias.

22. Give two examples of how a probability experiment could accidentally become biased.

23. A basketball player makes 36 of 50 free throws. Calculate the experimental probability of success.

24. Using the probability from Question 23, predict approximately how many successful shots the player might make out of 200 similar attempts.

25. Explain why your answer to Question 24 is not guaranteed.

26. A factory finds 12 defective products in a sample of 600. Calculate the experimental probability that a product is defective.

27. Use the result from Question 26 to predict approximately how many defective products might occur in 5,000 products if conditions remain similar.

28. Why should scientists be cautious when using an experimental probability calculated from only a few trials?

29. Five groups each flip a coin 100 times and obtain:

47, 54, 49, 52, and 51 heads.

Calculate the combined experimental probability of heads.

30. Challenge: A student claims a six-sided die is unfair. The student rolls it 20 times and obtains six 6s.

Calculate the experimental probability of rolling a 6 and compare it with the theoretical probability. Then explain why 20 trials alone may not provide enough evidence to decide whether the die is unfair and describe an improved investigation that could provide stronger evidence.

Theoretical Probability

Theoretical probability describes what we expect to happen based on mathematics and the possible outcomes of an event.

Unlike experimental probability, we do not need to perform an experiment first.

For example, consider a fair six-sided die.

The possible outcomes are:

1, 2, 3, 4, 5, 6

Because the die is fair, each outcome is equally likely.

There is one way to roll a 6 out of six possible outcomes.

Therefore:

P(6) = 1/6

This is a theoretical probability because it is calculated from what we know about the possible outcomes rather than from observed experimental results.

Theoretical probability allows us to answer questions such as:

• What is the probability of flipping heads?
• What is the probability of rolling an even number?
• What is the probability of drawing a heart from a deck of cards?
• Which outcome is more likely?
• How many times should we expect an event to occur?

What Is Theoretical Probability?

Theoretical probability is the probability of an event calculated using mathematical reasoning about all possible outcomes.

When all outcomes are equally likely:

P(event) = number of favourable outcomes / total number of possible outcomes

For example, a fair coin has two possible outcomes:

Heads, Tails

There is one favourable outcome for heads.

Therefore:

P(heads) = 1/2

= 0.5

= 50%

No coin needs to be flipped to calculate this probability.

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Possible Outcomes

An outcome is one possible result of a probability experiment.

For a coin:

Heads or Tails

For a standard six-sided die:

1, 2, 3, 4, 5, 6

For a spinner labelled A, B, C and D:

A, B, C, D

Before calculating a theoretical probability, we need to identify all possible outcomes.

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The Sample Space

The sample space is the complete set of possible outcomes.

It is often written inside braces.

For a coin:

S = {H, T}

For a six-sided die:

S = {1, 2, 3, 4, 5, 6}

For a spinner with four numbered sections:

S = {1, 2, 3, 4}

Identifying the sample space helps ensure that we count every possible outcome.

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Favourable Outcomes

A favourable outcome is an outcome that satisfies the event we are investigating.

Suppose a die is rolled and we want:

P(rolling an even number)

Sample space:

{1, 2, 3, 4, 5, 6}

Favourable outcomes:

{2, 4, 6}

There are:

3 favourable outcomes

out of:

6 possible outcomes

Therefore:

P(even) = 3/6

= 1/2

= 0.5

= 50%

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Equally Likely Outcomes

The basic theoretical probability formula works directly when all individual outcomes are equally likely.

For a fair die:

P(1) = P(2) = P(3) = P(4) = P(5) = P(6)

Each probability is:

1/6

For a fair coin:

P(heads) = P(tails)

Each probability is:

1/2

The word fair is important because it tells us that the relevant outcomes are equally likely.

When Outcomes Are Not Equally Likely

We cannot always simply count outcomes.

Imagine a spinner where:

• half the spinner is red
• one quarter is blue
• one quarter is green

There are three colour outcomes:

red, blue, green

But this does not mean:

P(red) = 1/3

The sections are not equal.

Instead:

P(red) = 1/2

P(blue) = 1/4

P(green) = 1/4

The probability depends on the size of each section.

This is why checking the assumptions of a probability model is important.

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Theoretical Probability Formula

For equally likely outcomes:

P(event) = favourable outcomes / total possible outcomes

Suppose we roll a standard die.

What is the probability of rolling a number greater than 4?

Possible outcomes:

1, 2, 3, 4, 5, 6

Favourable outcomes:

5, 6

Therefore:

P(number > 4) = 2/6

= 1/3

≈ 0.333

≈ 33.3%

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Worked Example: Rolling a 3

A fair six-sided die is rolled.

Sample space:

{1, 2, 3, 4, 5, 6}

Favourable outcome:

{3}

Therefore:

P(3) = 1/6

≈ 0.167

≈ 16.7%

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Worked Example: Rolling an Odd Number

Odd numbers on a die are:

1, 3, 5

Therefore:

P(odd) = 3/6

= 1/2

= 50%

Rolling an odd number has an even chance.

---

Worked Example: Rolling Less Than 5

Numbers less than 5 are:

1, 2, 3, 4

There are four favourable outcomes.

Therefore:

P(number < 5) = 4/6

= 2/3

≈ 66.7%

This event is more likely to happen than not happen.

Impossible Events

An impossible event has no favourable outcomes.

For example:

What is the probability of rolling an 8 on a standard six-sided die?

There are:

0 favourable outcomes

out of:

6 possible outcomes

Therefore:

P(8) = 0/6

= 0

An impossible event has probability:

0 = 0%

---

Certain Events

A certain event includes every possible outcome.

For example:

What is the probability of rolling a number less than 7?

Possible favourable outcomes:

1, 2, 3, 4, 5, 6

Therefore:

P(number < 7) = 6/6

= 1

= 100%

A certain event has probability:

1

---

Probability Can Be Written in Different Forms

Suppose:

P(event) = 3/4

As a decimal:

3 ÷ 4 = 0.75

As a percentage:

0.75 × 100 = 75%

Therefore:

3/4 = 0.75 = 75%

All three forms describe the same probability.

---

Simplifying Probability Fractions

Probability fractions should usually be simplified.

Suppose:

P(even number on a die) = 3/6

Divide numerator and denominator by 3:

3/6 = 1/2

Both are mathematically correct, but:

1/2

is the simplest form.

---

Probability with Coloured Counters

A bag contains:

5 red counters

3 blue counters

2 green counters

Total:

5 + 3 + 2 = 10 counters

Assume each counter has an equal chance of being selected.

Then:

P(red) = 5/10 = 1/2

P(blue) = 3/10

P(green) = 2/10 = 1/5

Therefore:

red is the most likely colour

and:

green is the least likely colour

Probability with Spinners

Suppose a spinner has eight equal sections.

They are:

3 red

2 blue

2 green

1 yellow

Because all sections are equal:

P(red) = 3/8

P(blue) = 2/8 = 1/4

P(green) = 2/8 = 1/4

P(yellow) = 1/8

Notice:

P(blue) = P(green)

Therefore, blue and green are equally likely.

Red is the most likely.

Yellow is the least likely.

---

The Importance of Equal Sections

Suppose another spinner has four sections labelled:

A, B, C, D

It might be tempting to say:

P(A) = 1/4

But we cannot know this simply from the four labels.

If the sections have different sizes, their probabilities are different.

The assumption:

"Each outcome has probability 1/4"

is only justified if the sections are equal or we otherwise know that the outcomes are equally likely.

Probability with Playing Cards

A standard deck contains:

52 cards

There are four suits:

• hearts
• diamonds
• clubs
• spades

Each suit contains:

13 cards

Therefore:

P = 13/52

= 1/4

= 25%

There are:

26 red cards

Therefore:

P(red card) = 26/52

= 1/2

= 50%

---

Worked Example: Drawing an Ace

A standard deck contains four aces.

Therefore:

P(ace) = 4/52

= 1/13

≈ 7.7%

---

Worked Example: Drawing a Face Card

The face cards are:

Jack, Queen, King

There are three face cards in each of four suits.

Number of face cards:

3 × 4 = 12

Therefore:

P(face card) = 12/52

= 3/13

≈ 23.1%

Complementary Events

The complement of an event is the event that it does not happen.

For example:

Event: rolling a 6

Complement: not rolling a 6

Since:

P(event) + P(not event) = 1

we can calculate:

P(not 6) = 1 − 1/6

= 5/6

This is often easier than counting all the other outcomes separately.

---

Worked Example: Complement

Suppose:

P(rain) = 0.35

Then:

P(no rain) = 1 − 0.35

= 0.65

= 65%

The probabilities of complementary events always total:

1

or:

100%

---

Probability of "Not"

The complement rule is particularly useful for questions containing:

not

For example:

A bag contains:

6 red
3 blue
1 green

Total = 10

P(red) = 6/10

Therefore:

P(not red) = 1 − 6/10

= 4/10

= 2/5

We could also count:

3 blue + 1 green = 4 non-red counters

which gives the same answer.

---

Comparing Theoretical Probabilities

Suppose:

P(A) = 1/5

P(B) = 3/5

Since:

3/5 > 1/5

Event B is more likely.

Probabilities allow us to compare events mathematically rather than relying only on words such as "likely" or "unlikely."

---

Worked Example: Which Is More Likely?

A fair die is rolled.

Event A

Rolling a number greater than 4.

Favourable outcomes:

5, 6

P(A) = 2/6 = 1/3

Event B

Rolling an even number.

Favourable outcomes:

2, 4, 6

P(B) = 3/6 = 1/2

Since:

1/2 > 1/3

Event B is more likely than Event A.

---

Expected Frequency

Theoretical probability can be used to predict approximately how often an event should occur.

Expected frequency = theoretical probability × number of trials

Suppose a fair coin is flipped:

200 times

P(heads) = 1/2

Expected heads:

200 × 1/2 = 100

We would expect approximately:

100 heads

This does not guarantee exactly 100 heads.

---

Worked Example: Expected Sixes

A fair die is rolled:

300 times

P(6) = 1/6

Expected number of sixes:

300 × 1/6

= 50

We would expect approximately 50 sixes.

The actual experimental result might be:

47, 52, 55, or another value.

---

Theoretical vs Experimental Probability

These two ideas are closely related but different.

Theoretical Probability

Based on:

mathematical reasoning

Example:

For a fair coin:

P(heads) = 0.5

Experimental Probability

Based on:

observed results

Suppose 100 flips produce 56 heads:

P(heads) = 56/100

= 0.56

The two probabilities are not exactly the same.

That is normal.

---

Comparing Theoretical and Experimental Results

Suppose a fair die is rolled 120 times.

Theoretically:

P(6) = 1/6

≈ 0.167

Expected sixes:

120 × 1/6 = 20

Suppose the experiment produces:

23 sixes

Experimental probability:

23/120

≈ 0.192

Theoretical probability:

0.167

Experimental probability:

0.192

The experimental result is slightly higher.

This difference can occur because of random variation.

---

More Trials Usually Improve the Comparison

Imagine repeatedly flipping a fair coin.

The theoretical probability remains:

0.5

As the number of trials increases, the experimental probability often becomes closer to the theoretical probability.

This is related to the law of large numbers.

Assumptions in Theoretical Probability

Theoretical probability depends on a mathematical model.

That model often requires assumptions.

For example, when we say:

P(6) = 1/6

we are assuming:

• the die has six possible faces
• the die is fair
• each face is equally likely
• the die is rolled normally
• no outside factor systematically favours one face

If these assumptions are incorrect, the theoretical model may not describe the real situation accurately.

---

Assumption: A Fair Coin

When we write:

P(heads) = 1/2

we usually assume the coin is fair.

But imagine a specially manufactured coin that lands heads much more frequently.

The mathematical model:

P(heads) = 1/2

might no longer be appropriate.

This shows why theoretical probability depends on the assumptions behind the model.

---

Assumption: A Fair Die

For a standard fair die:

P(each face) = 1/6

But if the die is weighted, damaged, or shaped unevenly, the outcomes may not be equally likely.

In that situation, experimental data might provide useful evidence about the actual probabilities.

---

Assumption: Random Selection

Suppose a bag contains:

4 red counters
6 blue counters

We calculate:

P(red) = 4/10

= 0.4

This assumes that every counter has an equal chance of being selected.

If someone looks into the bag and deliberately chooses a colour, the selection is no longer random.

The theoretical model no longer describes the process properly.

---

Assumption: Replacement

Suppose a bag contains:

5 red counters
5 blue counters

Initially:

P(red) = 5/10

= 1/2

Suppose a red counter is selected and not replaced.

The bag now contains:

4 red
5 blue

Total = 9

The probability of red on the next selection becomes:

P(red) = 4/9

The probabilities changed because the contents changed.

When objects are replaced after each selection, probabilities can remain constant from trial to trial.

---

Models Are Simplifications

A probability model is a mathematical description of a random situation.

Models are useful because they allow us to:

• calculate probabilities
• compare outcomes
• make predictions
• estimate expected frequencies

But models depend on assumptions.

For example:

"A coin has a 50% chance of heads."

is a model based on the assumption of a fair coin.

Real objects and real situations may not perfectly match ideal mathematical models.

---

When Experimental Results Disagree Strongly

Suppose a die is theoretically fair.

P(6) = 1/6

It is rolled:

6,000 times

Suppose 6 appears:

2,000 times

Experimental probability:

2000/6000

= 1/3

≈ 33.3%

This is very different from:

1/6 ≈ 16.7%

A large and persistent difference after many trials might suggest that we should investigate:

• whether the die is fair
• whether the experiment was biased
• whether the data were recorded correctly
• whether the theoretical assumptions are appropriate

Theoretical and experimental probability can therefore be used together to evaluate a probability model.

---

Theoretical Probability in Real Life

Many real-world probability problems are more complicated than coins and dice.

Theoretical models are used in:

Genetics

Predicting possible inheritance patterns.

Engineering

Estimating the probability of system failures.

Insurance

Modeling risks.

Games

Calculating possible outcomes and fairness.

Computer Science

Designing randomized algorithms and simulations.

Physics

Modeling processes involving uncertainty.

Weather and Climate Science

Probability models contribute to predictions about possible future conditions.

In real-world applications, theoretical models may be combined with large amounts of observed data.

---

Probability Trees

For situations involving several stages, a probability tree can help organize possible outcomes.

For example, flipping a coin twice produces:

First flip: H or T

Second flip: H or T

Possible combined outcomes:

HH

HT

TH

TT

There are four equally likely outcomes.

Therefore:

P(two heads) = 1/4

P(one head and one tail) = 2/4 = 1/2

P(two tails) = 1/4

Listing Outcomes Systematically

When a problem has several stages, listing outcomes systematically prevents us from missing possibilities.

Suppose a coin is flipped and a die is rolled.

Coin outcomes:

H, T

Die outcomes:

1, 2, 3, 4, 5, 6

Combined outcomes:

H1, H2, H3, H4, H5, H6
T1, T2, T3, T4, T5, T6

Total outcomes:

12

Suppose we want:

P(heads and an even number)

Favourable outcomes:

H2, H4, H6

Therefore:

P(heads and even) = 3/12

= 1/4

---

Checking Your Answer

A theoretical probability should always satisfy:

0 ≤ P(event) ≤ 1

If you calculate:

P(event) = 1.4

something has gone wrong.

Similarly:

P(event) = −0.2

cannot be correct.

Probability can never be:

• negative
• greater than 1
• less than 0%
• greater than 100%

---

A Strategy for Theoretical Probability Problems

When solving a probability problem:

Identify the experiment

What is happening?

Identify the sample space

What are all possible outcomes?

Check whether outcomes are equally likely

Do not assume this automatically.

Identify favourable outcomes

Which outcomes satisfy the event?

Calculate

For equally likely outcomes:

P(event) = favourable outcomes / total outcomes

Simplify

Simplify fractions where appropriate.

Convert if needed

Express the answer as a fraction, decimal, or percentage.

Check

Make sure:

0 ≤ P ≤ 1

Interpret

Explain what the probability means in context.

---

Worked Example: Full Solution

A bag contains:

7 red counters

5 blue counters

3 green counters

One counter is selected randomly.

Total outcomes

7 + 5 + 3 = 15 counters

Probability of red

P(red) = 7/15

≈ 46.7%

Probability of blue

P(blue) = 5/15

= 1/3

≈ 33.3%

Probability of green

P(green) = 3/15

= 1/5

= 20%

Comparison

Red is the most likely colour.

Green is the least likely.

Probability of not green

P(not green) = 1 − 1/5

= 4/5

= 80%

---

Worked Example: Comparing Theory and Experiment

A fair spinner has five equal sections:

A, B, C, D, E

Theoretical probability of A:

P(A) = 1/5

= 0.20

The spinner is spun 100 times.

A occurs 23 times.

Experimental probability:

P(A) = 23/100

= 0.23

Comparison:

The theoretical probability is 0.20.

The experimental probability is 0.23.

The experimental result is slightly higher, which can reasonably occur because of random variation.

More trials would generally provide a more stable comparison.

---

Common Mistakes

Using the formula when outcomes are not equally likely

Counting outcomes works directly only when each individual outcome has equal probability.

Forgetting some possible outcomes

Write the sample space when necessary.

Counting favourable outcomes incorrectly

Check exactly what the question asks.

Assuming "fair" and "random" mean the same thing

Random means the result cannot be known with certainty beforehand.

Fair means equivalent outcomes have equal probabilities.

Confusing theoretical and experimental probability

Theoretical probability comes from a mathematical model.

Experimental probability comes from observations.

Expecting theory and experiment to match exactly

Random variation is normal.

Forgetting to simplify fractions

For example:

3/6 = 1/2

Giving a probability greater than 1

Probabilities must lie between 0 and 1.

Assuming a model without stating its assumptions

For example, P(6) = 1/6 assumes a fair six-sided die.

Treating expected frequency as a guarantee

Expected frequency describes what we anticipate, not exactly what must happen.

Ignoring whether objects are replaced

Without replacement, probabilities can change between selections.

---

Key Terms

Theoretical probability — Probability calculated mathematically from possible outcomes and an assumed probability model.

Experimental probability — Probability calculated from observed results.

Probability model — A mathematical representation of a random situation.

Outcome — One possible result.

Event — An outcome or collection of outcomes being investigated.

Sample space — The complete set of possible outcomes.

Favourable outcome — An outcome that satisfies the event being investigated.

Equally likely — Having the same probability of occurring.

Fair — Describing a probability device in which equivalent outcomes have equal probabilities.

Random — Describing a process whose individual outcome cannot be predicted with certainty.

Impossible event — An event with probability 0.

Certain event — An event with probability 1.

Complement — The event that the original event does not occur.

Complementary events — An event and its complement; their probabilities total 1.

Expected frequency — The number of times an event is predicted to occur based on theoretical probability.

Trial — One repetition of a probability experiment.

Relative frequency — The proportion of trials in which an event occurs.

Random variation — Natural differences between observed results caused by randomness.

Assumption — A condition accepted as true when constructing or using a mathematical model.

Replacement — Returning a selected object before another selection so the original probabilities are restored.

Without replacement — Selecting objects without returning them, causing probabilities to change.

Bias — A systematic influence that favours certain outcomes.

Simulation — A model used to imitate a random process.

Probability tree — A diagram used to organize outcomes and probabilities across several stages.

Expected outcome — A result anticipated from a theoretical model.

---

Key Takeaways

• Theoretical probability is calculated using a mathematical model rather than observed experimental results.
• For equally likely outcomes:

P(event) = favourable outcomes / total possible outcomes

• The sample space contains all possible outcomes.
• Favourable outcomes are the outcomes that satisfy the event being investigated.
• The simple counting formula assumes that the individual outcomes are equally likely.
• Fair coins, fair dice, and equal spinner sections are common examples of equally likely outcomes.
• An impossible event has probability 0.
• A certain event has probability 1.
• Probabilities can be expressed as fractions, decimals, or percentages.
• Probabilities must always lie between 0 and 1.
• Complementary events have probabilities that total 1.
• The complement rule is:

P(not A) = 1 − P(A)

• Theoretical probability can be used to calculate expected frequencies.
• Expected frequency is:

theoretical probability × number of trials

• Expected frequency is not a guaranteed result.
• Theoretical probability and experimental probability may differ.
• Random variation causes experimental results to fluctuate.
• With many trials, experimental probability often becomes closer to theoretical probability.
• Theoretical probability depends on assumptions.
• Common assumptions include fairness, equal likelihood, random selection, and consistent experimental conditions.
• If those assumptions are incorrect, the theoretical model may not accurately represent reality.
• Strong and persistent differences between theoretical and experimental results may suggest that the model or experiment should be investigated.
• Replacement can keep probabilities constant between repeated selections.
• Without replacement, probabilities may change after each selection.
• Probability trees and systematic lists help organize multi-stage experiments.
• Probability models are simplifications of real situations and should always be interpreted with their assumptions in mind.

When solving a theoretical probability problem, remember:

Outcomes → Assumptions → Favourable Outcomes → Calculate → Check → Interpret

Always ask:

What are all the possible outcomes?

Are they equally likely?

Which outcomes satisfy the event?

What assumptions am I making?

Does my probability fall between 0 and 1?

How would I expect experimental results to compare with this theoretical value?

---

Check Your Understanding

1. Define theoretical probability.

2. Write the formula for theoretical probability when outcomes are equally likely.

3. What is a sample space?

4. Write the sample space for a standard six-sided die.

5. Calculate the probability of rolling a 5.

6. Calculate the probability of rolling an even number.

7. Calculate the probability of rolling a number greater than 2.

8. Calculate the probability of rolling a 9.

9. Calculate the probability of rolling a number less than 7.

10. A bag contains 5 red, 4 blue, and 1 green counter. Calculate:

a. P(red)
b. P(blue)
c. P(green)
d. P(not red)

11. Which colour in Question 10 is most likely to be selected?

12. A spinner has eight equal sections: three red, two blue, two green, and one yellow. Calculate the theoretical probability of each colour.

13. Which colours in Question 12 are equally likely?

14. Explain why simply counting the number of colours would give incorrect probabilities if the spinner sections were different sizes.

15. A standard deck contains 52 cards. Calculate:

a. P
b. P(red card)
c. P(ace)

16. A fair coin is flipped 500 times. How many heads would you expect?

17. A fair die is rolled 600 times. How many sixes would you expect?

18. Explain why the actual number in Question 17 might not be exactly equal to the expected number.

19. A fair die is rolled 120 times and produces 24 sixes. Calculate the experimental probability of rolling a 6 and compare it with the theoretical probability.

20. Does the result in Question 19 prove that the die is unfair? Explain.

21. What assumptions are made when we say that P(heads) = 1/2 for a coin?

22. Explain why a weighted die may not satisfy the usual theoretical probability model.

23. A bag contains equal-sized counters, but a student looks inside before selecting one. Which assumption of the theoretical model has been violated?

24. A bag contains 5 red and 5 blue counters. One red counter is selected and not replaced. What is the probability of selecting red on the next draw?

25. Explain why probabilities can change when objects are selected without replacement.

26. A fair coin is flipped twice. List all possible outcomes.

27. Using your answer to Question 26, calculate:

a. P(two heads)
b. P(exactly one head)
c. P(no heads)

28. A coin is flipped and a six-sided die is rolled. How many combined outcomes are possible?

29. Calculate the probability of obtaining tails and an odd number in Question 28.

30. Challenge: A game uses a spinner divided into 12 equal sections. Five sections are red, four are blue, two are green, and one is yellow.

a. Calculate the theoretical probability of each colour.
b. Predict the expected frequency of each colour in 600 spins.
c. An experiment produces 261 red, 188 blue, 101 green, and 50 yellow results. Calculate the experimental probability of each colour.
d. Compare the theoretical and experimental probabilities.
e. Explain why the results are not expected to match the theoretical probabilities exactly.
f. Identify the assumptions that must be true for the theoretical model to be appropriate.

Compound Events

Many probability situations involve more than one event.

Instead of asking:

"What is the probability of rolling a 6?"

we might ask:

"What is the probability of rolling a 6 and then rolling an even number?"

Instead of asking:

"What is the probability of flipping heads?"

we might ask:

"What is the probability of flipping two heads in a row?"

These are examples of compound events.

A compound event combines two or more events, and we can use organized lists, outcome diagrams, probability trees, and multiplication rules to determine their probabilities.

Compound probability is important because many real situations involve sequences or combinations of events rather than a single outcome.

What Is a Compound Event?

A compound event consists of two or more simple events.

For example:

Event 1: Flip a coin.

Event 2: Roll a die.

A possible compound outcome is:

Heads and 4

Another example is flipping a coin twice.

Possible compound outcomes are:

HH, HT, TH, TT

Each result contains outcomes from two events.

---

Simple Events and Compound Events

A simple event involves one outcome or one stage.

Example:

Rolling a 5

A compound event combines events or outcomes.

Examples:

• rolling a 5 and then a 3
• flipping heads twice
• selecting a red counter and then a blue counter
• rolling an even number or a number greater than 4
• selecting two aces from a deck

Compound events can involve:

AND

or:

OR

Understanding the wording is extremely important.

---

Sample Spaces for Compound Events

When several events occur, the sample space includes all possible combined outcomes.

Suppose a coin is flipped twice.

First flip:

H or T

Second flip:

H or T

The complete sample space is:

{HH, HT, TH, TT}

There are:

4 possible outcomes

If the coin is fair, each outcome has probability:

1/4

---

Organized Outcome Diagrams

One way to display compound outcomes is with an outcome diagram.

For two coin flips:

An organized diagram helps prevent us from:

• forgetting outcomes
• counting outcomes twice
• incorrectly assuming how many possibilities exist

Worked Example: Two Coin Flips

Sample space:

{HH, HT, TH, TT}

Probability of two heads

Only:

HH

is favourable.

Therefore:

P(two heads) = 1/4

= 25%

Probability of exactly one head

Favourable outcomes:

HT, TH

Therefore:

P(exactly one head) = 2/4

= 1/2

= 50%

Probability of at least one head

Favourable outcomes:

HH, HT, TH

Therefore:

P(at least one head) = 3/4

= 75%

---

"Exactly" and "At Least"

Probability questions often use important words.

Exactly one head

Means:

one head and no more

For two flips:

HT, TH

At least one head

Means:

one or more heads

For two flips:

HH, HT, TH

At most one head

Means:

zero or one head

For two flips:

HT, TH, TT

Carefully reading these words can completely change the answer.

---

Probability Trees

A probability tree is a diagram showing possible outcomes at each stage of a compound event.

For a fair coin flipped twice:

First flip:

• H → 1/2
• T → 1/2

From each branch, the second flip has:

• H → 1/2
• T → 1/2

This creates four paths:

H → H

H → T

T → H

T → T

Each complete path represents one possible compound outcome.

Multiplying Along a Probability Tree

To calculate the probability of following a particular path:

Multiply the probabilities along the branches.

For:

Heads followed by Heads

P(HH) = 1/2 × 1/2

= 1/4

For:

Heads followed by Tails

P(HT) = 1/2 × 1/2

= 1/4

The same applies to TH and TT.

This gives:

Check:

1/4 + 1/4 + 1/4 + 1/4 = 1

All possible outcomes together must have a total probability of 1.

---

The Multiplication Rule

When calculating the probability that one event occurs AND another event occurs:

Multiply probabilities along the path.

For independent events:

P(A and B) = P(A) × P(B)

For example:

P(heads and then rolling 6)

= 1/2 × 1/6

= 1/12

This rule becomes especially useful when many outcomes are possible.

---

Independent Events

Two events are independent when the outcome of one event does not affect the probability of the other.

Examples include:

• flipping a coin and rolling a die
• rolling a die twice
• flipping a coin several times
• spinning a spinner twice

Suppose you flip heads on the first coin toss.

Does this change the probability of heads on the second toss?

No.

The second flip still has:

P(heads) = 1/2

The events are independent.

---

Independent Does Not Mean Different

Suppose you roll a die twice.

The first roll is:

6

The probability of rolling another 6 is still:

1/6

The die does not "remember" the previous result.

Similarly, if a coin produces five heads in a row, then for a fair coin:

P(heads on next flip) = 1/2

and:

P(tails on next flip) = 1/2

The previous results do not change the probabilities.

---

Worked Example: Coin and Die

A fair coin is flipped and a fair six-sided die is rolled.

What is the probability of:

Heads AND 6?

P(heads) = 1/2

P(6) = 1/6

The events are independent.

Therefore:

P(heads and 6)

= 1/2 × 1/6

= 1/12

≈ 8.3%

Outcome Diagram: Coin and Die

We can also list the complete sample space.

There are:

12 equally likely outcomes

Only one is:

H6

Therefore:

P(H and 6) = 1/12

The outcome diagram and multiplication method produce the same result.

---

Worked Example: Heads and an Even Number

Using the same experiment:

P(heads) = 1/2

P(even number) = 3/6 = 1/2

Therefore:

P(heads and even)

= 1/2 × 1/2

= 1/4

Looking at the outcome diagram, the favourable outcomes are:

H2, H4, H6

There are:

3 favourable outcomes out of 12.

Therefore:

3/12 = 1/4

Again, both methods agree.

---

Three Independent Events

The multiplication rule can be extended.

Suppose a fair coin is flipped three times.

What is the probability of:

HHH?

P(HHH)

= 1/2 × 1/2 × 1/2

= 1/8

The complete sample space is:

HHH

HHT

HTH

HTT

THH

THT

TTH

TTT

There are eight equally likely outcomes.

---

Probability of a Specific Sequence

Suppose a fair coin is flipped four times.

What is:

P(HTTH)?

Multiply:

1/2 × 1/2 × 1/2 × 1/2

= 1/16

Any particular sequence of four fair coin flips has probability:

1/16

This includes:

HHHH
HTTH
TTTT
THHT

Each specific sequence is equally likely.

---

Dependent Events

Two events are dependent when the outcome of one event changes the probability of another.

A common example involves selecting objects without replacement.

Suppose a bag contains:

3 red counters

2 blue counters

Total:

5 counters

We select one counter and do not replace it.

If the first counter is red, the bag changes.

Before the first selection:

P(red) = 3/5

After selecting red:

There are now:

2 red

2 blue

Total:

4 counters

Therefore:

P(red on second draw | red first) = 2/4

= 1/2

The first event changed the probability of the second event.

The events are dependent.

With Replacement vs Without Replacement

This distinction is extremely important.

With Replacement

The selected object is returned before the next selection.

The original probabilities are restored.

Events are often independent.

Without Replacement

The selected object is not returned.

The contents change.

The probabilities for later selections change.

Events are dependent.

---

Worked Example: Without Replacement

A bag contains:

4 red counters

3 blue counters

Two counters are selected without replacement.

Find:

P(red then red)

First selection:

P(red) = 4/7

After selecting red:

3 red counters remain.

Total counters remaining:

6

Therefore:

P(second red | first red) = 3/6

Now multiply:

P(red then red)

= 4/7 × 3/6

= 12/42

= 2/7

≈ 28.6%

Notice that we did not calculate:

4/7 × 4/7

because the first selection changed the contents of the bag.

---

Worked Example: Red Then Blue

Using the same bag:

4 red
3 blue

Total = 7

Find:

P(red then blue)

First:

P(red) = 4/7

After selecting red:

There are still:

3 blue

but only:

6 counters total

Therefore:

P(blue second | red first) = 3/6

Multiply:

P(red then blue)

= 4/7 × 3/6

= 12/42

= 2/7

---

The Probability Tree Changes

For dependent events, the probabilities on the second set of branches depend on what happened first.

Suppose a bag contains:

2 red counters
1 blue counter

Two counters are selected without replacement.

First draw:

P(R) = 2/3

P(B) = 1/3

If the first counter is red:

Remaining:

1 red, 1 blue

Therefore:

P(R second) = 1/2

P(B second) = 1/2

If the first counter is blue:

Remaining:

2 red, 0 blue

Therefore:

P(R second) = 1

P(B second) = 0

This changing probability is what makes the events dependent.

Conditional Probability

When one event affects another, we often describe the probability using the word given.

For example:

P(red second given red first)

means:

the probability of selecting red on the second draw given that red was selected first.

If a bag originally contains:

4 red and 3 blue

and a red counter has already been removed:

P(red second | red first)

= 3/6

= 1/2

The vertical line:

|

can be read as:

"given that"

This idea is called conditional probability.

---

Adding Probabilities

Probability trees also help with situations where several different paths satisfy the same event.

Suppose a fair coin is flipped twice.

What is the probability of getting:

exactly one head?

There are two possible paths:

HT

or:

TH

Each has probability:

1/2 × 1/2 = 1/4

Now add the probabilities:

P(exactly one head)

= 1/4 + 1/4

= 1/2

This gives us another important rule:

Multiply along branches.

Add between separate favourable paths.

---

AND Usually Means Multiply

Consider:

P(heads AND heads)

We follow one path:

H → H

Therefore:

1/2 × 1/2 = 1/4

The word AND often indicates that several events must occur together.

This usually involves multiplication.

---

OR Often Means Add

Consider:

P(HH OR TT)

P(HH) = 1/4

P(TT) = 1/4

These are separate outcomes.

Therefore:

P(HH or TT)

= 1/4 + 1/4

= 1/2

The word OR often involves adding probabilities of different favourable outcomes.

However, care is needed when events overlap.

---

Mutually Exclusive Events

Events are mutually exclusive when they cannot occur at the same time in a single trial.

For one die roll:

rolling a 2

and:

rolling a 5

are mutually exclusive.

You cannot roll both on the same die at the same time.

Therefore:

P(2 or 5)

= P(2) + P(5)

= 1/6 + 1/6

= 2/6

= 1/3

---

Events That Overlap

Some events are not mutually exclusive.

Consider rolling one die.

Event A:

rolling an even number

= {2, 4, 6}

Event B:

rolling a number greater than 3

= {4, 5, 6}

Notice:

4 and 6

belong to both events.

If we simply add:

3/6 + 3/6

we would count 4 and 6 twice.

The correct favourable outcomes are:

{2, 4, 5, 6}

Therefore:

P(even OR greater than 3)

= 4/6

= 2/3

For overlapping events, we must avoid double counting.

---

General Addition Rule

For two events A and B:

P(A or B) = P(A) + P(B) − P(A and B)

We subtract the overlap because it was counted twice.

Using the previous example:

P(even) = 3/6

P(greater than 3) = 3/6

P(even and greater than 3) = 2/6

Therefore:

P(even or greater than 3)

= 3/6 + 3/6 − 2/6

= 4/6

= 2/3

---

"At Least One"

Compound probability questions frequently ask for:

at least one

For example:

What is the probability of getting at least one head in three coin flips?

We could count:

HHH
HHT
HTH
HTT
THH
THT
TTH

That works, but there is an easier method.

The only outcome that does not contain a head is:

TTT

P(TTT)

= 1/2 × 1/2 × 1/2

= 1/8

Therefore:

P(at least one head)

= 1 − 1/8

= 7/8

This is called using the complement.

---

The Complement Strategy

Sometimes it is easier to calculate what we do not want.

The general rule is:

P(at least one success) = 1 − P(no successes)

For example, suppose a die is rolled twice.

What is the probability of rolling at least one 6?

Instead of listing every favourable outcome, calculate:

P(no 6 on one roll) = 5/6

P(no 6 on either roll)

= 5/6 × 5/6

= 25/36

Therefore:

P(at least one 6)

= 1 − 25/36

= 11/36

≈ 30.6%

---

Worked Example: At Least One Head

A fair coin is flipped four times.

Find:

P(at least one head)

Probability of tails on one flip:

1/2

Probability of four tails:

P(TTTT)

= (1/2)⁴

= 1/16

Therefore:

P(at least one head)

= 1 − 1/16

= 15/16

= 93.75%

---

Compound Events with Spinners

Suppose a spinner has four equal sections:

Red, Blue, Green, Yellow

It is spun twice.

What is:

P(red then blue)?

P(red) = 1/4

P(blue) = 1/4

The spins are independent.

Therefore:

P(red then blue)

= 1/4 × 1/4

= 1/16

Worked Example: Same Colour Twice

Suppose a spinner has three equal sections:

Red, Blue, Green

It is spun twice.

Possible same-colour outcomes:

RR

BB

GG

Each has probability:

1/3 × 1/3 = 1/9

There are three favourable paths.

Therefore:

P(same colour)

= 1/9 + 1/9 + 1/9

= 3/9

= 1/3

---

Compound Events with Cards

Suppose a standard 52-card deck is used.

Two cards are drawn without replacement.

Find the probability that both cards are aces.

First card:

P(ace) = 4/52

After drawing an ace:

3 aces remain.

51 cards remain.

Therefore:

P(second ace | first ace) = 3/51

Multiply:

P(two aces)

= 4/52 × 3/51

= 12/2652

= 1/221

≈ 0.45%

The events are dependent because the first card is not replaced.

---

What If the Card Is Replaced?

Suppose the first card is returned to the deck and the deck is shuffled.

Then:

P(ace first) = 4/52

and:

P(ace second) = 4/52

The probabilities remain the same.

Therefore:

P(two aces with replacement)

= 4/52 × 4/52

= 1/13 × 1/13

= 1/169

The replacement changes the probability structure.

---

Reading a Probability Tree

A probability tree can tell us:

• possible outcomes
• probability at each stage
• probability of each complete path
• whether probabilities change
• whether events appear independent or dependent

When reading a tree:

Step 1: Start at the beginning.

Step 2: Follow one complete path.

Step 3: Multiply probabilities along that path.

Step 4: Repeat for other favourable paths.

Step 5: Add the probabilities of the favourable paths.

This process works for many compound probability problems.

---

Checking a Probability Tree

At every branching point, probabilities should total:

1

For example:

P(H) + P(T)

= 1/2 + 1/2

= 1

For a bag:

P(red) = 3/5

P(blue) = 2/5

Then:

3/5 + 2/5 = 1

If branch probabilities do not total 1, check your tree.

---

Independent or Dependent?

Ask one simple question:

Does the first event change the probability of the second event?

If no:

The events are independent.

If yes:

The events are dependent.

Examples:

---

Worked Example: Identifying Dependence

A bag contains:

5 red counters
3 blue counters

A counter is selected.

Situation A

The counter is replaced.

Second selection:

P(red) = 5/8

The probability has not changed.

Independent

Situation B

The counter is not replaced and the first counter was red.

Second selection:

P(red) = 4/7

The probability changed.

Dependent

---

Real-World Compound Events

Compound probabilities occur in many situations.

Genetics

Several genetic traits may be inherited together.

Manufacturing

A product may need several components to function correctly.

Reliability Engineering

A system may depend on several components working successfully.

Medicine

Several risk factors or test outcomes may be considered together.

Weather

Sequences or combinations of weather events may be analyzed.

Sports

Analysts may examine sequences of shots, passes, or scoring events.

Games

Cards, dice, spinners, and other random processes frequently involve compound events.

Probability trees help organize these more complicated situations.

---

Compound Probability and Experimental Results

Compound probabilities can also be investigated experimentally.

Suppose two coins are flipped 100 times.

The theoretical probability of HH is:

1/2 × 1/2 = 1/4

Therefore, expected HH outcomes:

100 × 1/4 = 25

Suppose the experiment produces:

28 HH outcomes

Experimental probability:

28/100 = 0.28

Theoretical probability:

0.25

The values are not identical, but the difference may simply be due to random variation.

---

Probability Trees as Models

A probability tree is a model.

It depends on correct assumptions.

For example, when constructing a tree for two coin flips, we may assume:

• the coin is fair
• the flips are independent

When constructing a tree for counters, we must know:

• how many counters of each colour exist
• whether selection is random
• whether counters are replaced
• whether all counters are equally likely to be selected

Incorrect assumptions produce incorrect probabilities.

---

Worked Example: Full Probability Tree Problem

A bag contains:

3 red counters

2 blue counters

Two counters are selected without replacement.

Find the probability of selecting one counter of each colour.

There are two favourable paths:

Red then Blue

or:

Blue then Red

Red then Blue

P(R then B)

= 3/5 × 2/4

= 6/20

= 3/10

Blue then Red

P(B then R)

= 2/5 × 3/4

= 6/20

= 3/10

Now add:

P(one of each)

= 3/10 + 3/10

= 6/10

= 3/5

= 60%

This demonstrates the key probability-tree strategy:

Multiply along each path.

Add the favourable paths.

---

Worked Example: Both the Same Colour

Using the same bag:

3 red
2 blue

Find:

P(both counters are the same colour)

Possible paths:

RR

or:

BB

RR

3/5 × 2/4

= 6/20

= 3/10

BB

2/5 × 1/4

= 2/20

= 1/10

Add:

3/10 + 1/10

= 4/10

= 2/5

= 40%

Notice:

P(one of each) + P(same colour)

= 3/5 + 2/5

= 1

These two events account for every possible result.

---

A Strategy for Compound Probability

When solving a compound probability problem:

Identify the events

What happens first, second, third, and so on?

Determine whether they are independent or dependent

Ask:

Does one event change the probability of another?

Organize the outcomes

Use:

• a list
• an outcome diagram
• a table
• a probability tree

Label probabilities

Check that probabilities at each branching point total 1.

Multiply along paths

Use multiplication when events occur together along one route.

Add favourable paths

If several paths satisfy the question, calculate each path and add them.

Consider the complement

For "at least one" questions, calculating the opposite event may be easier.

Check your answer

Probability must satisfy:

0 ≤ P ≤ 1

---

Common Mistakes

Adding when you should multiply

For events occurring along one path:

multiply

Multiplying when you should add

For separate favourable paths:

add

Assuming every event is independent

Without replacement often creates dependent events.

Forgetting to change the denominator

If an object is not replaced, the total number of objects decreases.

Forgetting to change the numerator

The number of favourable objects may also decrease.

Treating replacement and no replacement as identical

They produce different probabilities.

Assuming previous coin flips affect the next flip

Independent events do not "remember" previous outcomes.

Confusing exactly with at least

"Exactly one" and "at least one" describe different events.

Forgetting multiple possible orders

"One red and one blue" includes:

RB and BR

Double counting overlapping events

When events overlap, do not count shared outcomes twice.

Forgetting the complement strategy

"At least one" questions are often easier using:

1 − P(none)

Drawing incomplete probability trees

Every possible outcome at each stage should be represented.

Forgetting to check branch totals

Branches from the same point should total 1.

---

Key Terms

Compound event — An event involving two or more simple events or conditions.

Simple event — An event involving a single outcome or stage.

Sample space — The complete set of possible outcomes.

Outcome diagram — An organized representation of possible combined outcomes.

Probability tree — A branching diagram showing possible outcomes and their probabilities across multiple stages.

Branch — One path segment on a probability tree representing an outcome.

Path — A sequence of branches representing a complete compound outcome.

Independent events — Events where one outcome does not change the probability of another.

Dependent events — Events where one outcome changes the probability of another.

Conditional probability — The probability of an event given that another event has occurred.

Replacement — Returning a selected object before the next selection.

Without replacement — Keeping the selected object out, causing later probabilities to change.

Multiplication rule — A rule used to calculate the probability of events occurring together along a path.

Addition rule — A rule used to combine probabilities of different favourable outcomes.

Mutually exclusive events — Events that cannot occur simultaneously in a single trial.

Overlapping events — Events that share one or more outcomes.

Complement — The event that the original event does not occur.

At least — A phrase meaning the stated amount or more.

At most — A phrase meaning the stated amount or fewer.

Exactly — A phrase indicating precisely the stated number of occurrences.

Favourable outcome — An outcome satisfying the event being investigated.

Fair — Describes a probability process in which equivalent outcomes have equal probabilities.

Random — Describes an outcome that cannot be predicted with certainty beforehand.

Probability model — A mathematical representation of a random process.

Tree diagram — Another name commonly used for a probability tree.

Conditional notation — Notation such as P(B | A), meaning the probability of B given that A has occurred.

---

Key Takeaways

• A compound event combines two or more events.
• Compound outcomes can be organized using lists, tables, outcome diagrams, and probability trees.
• Probability trees show possible outcomes across several stages.
• Each complete route through a probability tree represents a possible compound outcome.
• Multiply probabilities along a path.
• Add probabilities of separate favourable paths.
• Independent events do not change each other's probabilities.
• Dependent events do change later probabilities.
• Repeated coin flips and die rolls are usually independent.
• Selecting objects without replacement creates dependent events.
• Replacement usually restores the original probabilities.
• For independent events:

P(A and B) = P(A) × P(B)

• Conditional probability describes the probability of one event given that another has already occurred.
• Probability trees for dependent events have changing branch probabilities.
• The word AND often indicates multiplication along a path.
• The word OR often involves adding probabilities of different outcomes.
• Mutually exclusive events cannot occur simultaneously.
• Overlapping events must not be counted twice.
• For overlapping events:

P(A or B) = P(A) + P(B) − P(A and B)

• "Exactly one" means one and only one.
• "At least one" means one or more.
• "At most one" means zero or one.
• "At least one" problems are often easiest using a complement.
• A useful rule is:

P(at least one success) = 1 − P(no successes)

• Branch probabilities leaving the same point should total 1.
• All complete outcomes together should have total probability 1.
• Probability trees depend on assumptions such as fairness, random selection, independence, and replacement.
• Compound theoretical probabilities can be compared with experimental results.
• Experimental results may differ from theoretical predictions because of random variation.

For probability trees, remember:

Multiply along. Add across.

For independence, ask:

Did the first event change the probability of the next event?

And for "at least one," ask:

Would it be easier to calculate none and subtract from 1?

---

Check Your Understanding

1. What is a compound event?

2. Explain the difference between independent and dependent events.

3. A fair coin is flipped twice. List the complete sample space.

4. Calculate the probability of:

a. HH
b. HT
c. exactly one head
d. at least one head
e. no heads

5. A fair coin is flipped three times. Calculate P(HHH).

6. Calculate the probability of exactly three tails in three flips.

7. Calculate the probability of at least one head in three flips.

8. Explain why using a complement makes Question 7 easier.

9. A fair coin is flipped and a fair die is rolled. Calculate:

a. P(heads and 6)
b. P(tails and an even number)
c. P(heads and a number greater than 4)

10. Are the coin and die events in Question 9 independent or dependent? Explain.

11. A bag contains 4 red and 3 blue counters. Two counters are selected without replacement. Calculate:

a. P(red then red)
b. P(red then blue)
c. P(blue then red)
d. P(blue then blue)

12. Check that the probabilities in Question 11 total 1.

13. Why do the probabilities change after the first selection in Question 11?

14. Repeat Question 11 assuming that the first counter is replaced before the second selection.

15. Explain why replacement changes the calculations.

16. A bag contains 3 red and 2 blue counters. Two are selected without replacement. Calculate the probability of selecting one of each colour.

17. Why must both RB and BR be included in Question 16?

18. A fair die is rolled twice. Calculate the probability of rolling two sixes.

19. Calculate the probability of rolling at least one 6 in two rolls.

20. A spinner has four equal sections. It is spun twice. Calculate the probability of landing on the same section both times.

21. Explain the difference between "exactly one success" and "at least one success."

22. A standard deck contains 52 cards. Two cards are drawn without replacement. Calculate the probability that both are aces.

23. Are the card selections in Question 22 independent or dependent? Explain.

24. How would the calculation change if the first card were replaced?

25. A die is rolled once. Find:

P(even OR greater than 3)

Explain why simply adding P(even) and P(greater than 3) would give an incorrect answer.

26. Explain the rule:

Multiply along branches and add favourable paths.

27. What should the probabilities leaving each branching point on a probability tree total?

28. Give two real-world situations involving compound events.

29. A fair coin is flipped four times. Calculate the probability of:

a. four tails
b. at least one head
c. the specific sequence HHTT

30. Challenge: A bag contains 5 red, 3 blue, and 2 green counters. Two counters are selected without replacement.

Construct a probability tree and use it to determine:

a. P(red then red)
b. P(red then blue)
c. P(blue then red)
d. P(two green counters)
e. P(one red and one blue, in either order)
f. P(at least one red counter)

Explain why the events are dependent and how the probabilities on the second branches change after the first counter is selected.

Probability in Everyday Life

Probability is not just something we calculate with coins, dice, cards, and spinners. Probability is used every day to describe uncertainty and help people make decisions.

Consider some common situations:

• A weather forecast reports a 70% chance of rain.
• A basketball player makes 82% of free throws.
• A doctor explains that a treatment is successful in a certain percentage of cases.
• An engineer estimates the probability that a component will fail.
• An insurance company estimates the risk of accidents.
• A student sees that there is only a small chance of winning a prize.
• A traveller considers the possibility that a flight could be delayed.

In each situation, probability helps answer an important question:

How likely is something to happen?

Probability does not tell us exactly what will happen. Instead, it helps us make decisions when the future is uncertain.

Probability Helps Us Deal with Uncertainty

Many decisions involve outcomes that cannot be predicted with complete certainty.

For example, suppose you are deciding whether to bring an umbrella.

The weather forecast says:

P(rain) = 0.80

or:

80% chance of rain

You do not know for certain that it will rain.

However, an 80% probability indicates that rain is considered likely under the forecast model.

You might therefore decide that carrying an umbrella is worthwhile.

Probability has helped you make a decision even though the outcome remains uncertain.

---

Probability in Weather Forecasting

Weather is one of the most familiar places where people encounter probability.

Forecasts may include statements such as:

20% chance of rain

50% chance of thunderstorms

80% chance of precipitation

These percentages describe the estimated probability of a defined weather event under the forecasting method being used.

A larger probability means the event is considered more likely.

For example:

The important idea is that probabilities describe likelihood, not guarantees.

What Does a 70% Chance of Rain Mean?

A common mistake is to think:

"70% chance of rain means it will rain for 70% of the day."

That is not generally what the percentage means.

It also does not simply mean:

"70% of the area will receive rain."

A probability of precipitation is intended to describe the likelihood of measurable precipitation at the relevant forecast location or area during the specified period, according to the forecasting system being used.

The exact definition can depend on the weather service and forecast product.

The key mathematical interpretation is:

70% represents a probability of 0.70 for the defined event.

It means the event is considered likely, but not certain.

---

Weather and Decision-Making

Suppose you are planning a picnic.

Forecast A

Probability of rain = 10%

You may decide the risk is small enough to continue.

Forecast B

Probability of rain = 50%

The decision becomes less obvious.

Forecast C

Probability of rain = 90%

You may decide to move the picnic indoors.

Notice that probability does not make the decision for you.

It provides information that can be used to make the decision.

Other factors matter too:

• Is shelter available?
• How important is the event?
• What happens if it rains?
• Can the event easily be moved?
• How reliable is the forecast?

---

Probability in Sports

Sports contain enormous amounts of probability data.

Examples include:

• shooting percentage
• batting statistics
• penalty success rates
• free-throw percentages
• save percentages
• scoring rates
• win/loss records

These statistics can help estimate the likelihood of future outcomes.

Suppose a basketball player attempts:

200 free throws

and makes:

160

Experimental probability of making a free throw:

P(success) = 160/200

= 0.80

= 80%

Based on these observations, we might estimate that the player's probability of making a similar future free throw is approximately 80%.

Sports Statistics Are Not Guarantees

Suppose a basketball player has an 80% free-throw success rate.

Does this mean:

the next shot will definitely go in?

No.

Does it mean:

exactly 8 of every 10 shots must go in?

No.

The player could:

• miss the next shot
• make ten shots in a row
• miss several shots in a row

An 80% success rate describes a pattern across many previous attempts.

Individual outcomes remain uncertain.

---

Worked Example: Sports Probability

A football player takes 50 penalty kicks and scores 39.

Experimental probability of scoring:

P(score) = 39/50

= 0.78

= 78%

Probability of not scoring based on these data:

P(not score) = 11/50

= 0.22

= 22%

If similar conditions continued for another 100 penalties, we might predict approximately:

100 × 0.78 = 78 goals

However, this is only an estimate.

---

Past Performance and Future Probability

Historical data can help estimate probabilities, but we should be careful.

A player's previous success rate may not perfectly predict future performance.

Conditions can change:

• opponent
• weather
• fatigue
• injury
• equipment
• playing surface
• skill development
• pressure
• strategy

Probability based on historical data is useful evidence, but it is not a guarantee about the future.

---

Probability and Risk

Risk involves the possibility that an unwanted event may occur.

Probability helps us describe how likely that event is.

For example:

• probability of an accident
• probability of flooding
• probability of equipment failure
• probability of a product being defective
• probability of a flight being delayed

Suppose a machine has a:

2% probability of failure during a particular operation

This can also be written as:

0.02

or approximately:

2 in 100

The probability is small, but it is not zero.

Probability and Consequence

Risk decisions should not consider probability alone.

We should also consider:

How serious would the consequences be?

Compare two situations.

Situation A

There is a 30% chance that you forget a pencil.

Situation B

There is a 1% chance that a critical safety system fails.

Situation B has the lower probability.

However, its consequences could be much more serious.

This means:

low probability does not necessarily mean unimportant.

When assessing risk, we often need to consider both:

likelihood

and:

consequence

---

Risk Matrices

Organizations sometimes use risk matrices to combine likelihood and consequence.

For example:

The exact categories used depend on the organization and situation.

A risk matrix does not replace careful analysis, but it can help people organize their thinking.

---

Everyday Safety Decisions

Imagine there is only a small probability of falling from a bicycle during one particular journey.

You might still wear a helmet because:

• falling is possible
• the consequences could be serious
• wearing a helmet is relatively easy

This demonstrates an important principle:

Decisions depend on both probability and consequences.

A sensible precaution can be worthwhile even when the probability of the unwanted event is relatively low.

---

Probability in Insurance

Insurance companies use probability and statistics to estimate risk.

They may analyze large amounts of historical data about events such as:

• vehicle accidents
• property damage
• theft
• storms
• equipment failure

Suppose historical data show that:

3 out of every 100 similar insured objects experience a particular type of loss each year.

Experimental estimate:

P(loss) = 3/100

= 0.03

= 3%

An insurer can combine probability information with the possible financial cost of the event when estimating risk.

For an individual person, however, a 3% probability does not tell us whether the event will happen to that specific person.

Probability in Transportation

Probability can help transportation systems analyze:

• delays
• accidents
• equipment failures
• traffic patterns
• passenger demand
• arrival times

Suppose a bus company records:

900 journeys

and:

765 arrive on time

Experimental probability of an on-time arrival:

765/900

= 0.85

= 85%

Based on these historical observations, a passenger might estimate that a similar journey has a reasonably high chance of arriving on time.

However:

85% does not mean guaranteed.

---

Worked Example: Travel Decision

Suppose you need to arrive at school before an important examination.

Bus A has historically arrived on time:

95% of the time

Bus B has historically arrived on time:

75% of the time

Probability provides useful information.

However, you might also consider:

• travel time
• departure time
• traffic
• cost
• alternative routes
• consequences of arriving late

Probability is therefore one part of a decision, not necessarily the entire decision.

---

Probability in Manufacturing

Factories use probability to monitor product quality.

Suppose a factory inspects:

2,000 products

and finds:

30 defective products

Experimental probability of a defect:

P(defect) = 30/2000

= 0.015

= 1.5%

If production conditions remain similar, this probability can help estimate the number of defects in future production.

For 10,000 products:

Expected defects:

10,000 × 0.015

= 150

This is an estimate, not a guarantee.

---

Probability and Quality Control

Suppose a factory normally has a defect rate of:

1%

A new sample of 1,000 products contains:

45 defects

Experimental defect rate:

45/1000 = 4.5%

This is considerably higher than the usual 1%.

The company might investigate:

• equipment
• materials
• manufacturing processes
• measurements
• worker procedures
• environmental conditions

Probability and statistics can help identify unusual patterns that deserve further investigation.

---

Probability in Medicine and Health Research

Probability is widely used in medical and health research.

Researchers may study:

• probability of recovery
• probability of side effects
• probability of developing a condition
• accuracy of medical tests
• effectiveness of treatments

Suppose a study reports that:

92 out of 100 participants experienced a particular outcome

The observed proportion is:

92/100 = 0.92

= 92%

However, interpreting medical probabilities can be more complicated than simple classroom probability.

Factors such as sample size, study design, population, and individual characteristics matter.

Relative Risk and Absolute Risk

Probability information can sometimes be presented in ways that sound more dramatic than they really are.

Consider a hypothetical example.

Suppose the probability of an event changes from:

1 in 1,000

to:

2 in 1,000

The probability has doubled.

But in absolute terms, it increased from:

0.1%

to:

0.2%

Both descriptions are mathematically related, but they communicate the information differently.

When interpreting risk information, ask:

What are the actual probabilities?

This helps put claims into context.

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Probability in Games

Games provide familiar examples of theoretical probability.

For a fair six-sided die:

P(rolling 6) = 1/6

≈ 16.7%

For a fair coin:

P(heads) = 1/2

= 50%

For a standard deck of 52 cards:

P(ace) = 4/52

= 1/13

≈ 7.7%

Understanding these probabilities helps us analyze whether games are fair and how likely different outcomes are.

Probability and Expected Outcomes

Suppose a game uses a fair six-sided die.

You win if you roll a 6.

Probability of winning:

P(win) = 1/6

Suppose you play:

600 times

Expected wins:

600 × 1/6

= 100

This does not mean exactly 100 wins must occur.

It means that approximately 100 wins would be expected according to the probability model.

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Probability and Financial Decisions

Probability is also used when evaluating uncertain financial outcomes.

For example, businesses may estimate:

• probability that a project finishes on time
• probability that demand exceeds supply
• probability that equipment fails
• probability that customers purchase a product

Suppose a company estimates:

P(high demand) = 0.65

This does not mean high demand will definitely occur.

It provides information that can be considered when deciding:

• how much inventory to produce
• how many workers are needed
• how much storage is required

Again:

Probability informs decisions. It does not eliminate uncertainty.

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Probability and Natural Hazards

Scientists study probabilities associated with natural hazards such as:

• floods
• earthquakes
• volcanic eruptions
• hurricanes and tropical cyclones
• droughts
• extreme temperatures

For example, a flood risk may be described using an annual probability.

An event with a:

1% annual probability

has a probability of:

0.01

during a particular year under the model being used.

A common mistake is to assume that if such an event happens this year, it cannot happen again for many years.

That is incorrect.

Probability describes likelihood, not a fixed timetable.

The "100-Year Flood" Misunderstanding

The phrase "100-year flood" can be misleading.

It does not mean:

one flood exactly every 100 years.

It generally refers to a flood magnitude associated with approximately:

1% annual probability

under the relevant statistical model.

That means such an event could occur:

• twice within a few years
• in consecutive years
• or not at all for a very long period

Probability does not create a schedule.

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Probability in Science

Scientists work with uncertainty constantly.

Probability helps researchers:

• analyze experimental data
• estimate uncertainty
• compare hypotheses
• study genetics
• model populations
• analyze radioactive decay
• predict possible outcomes
• assess whether patterns may have occurred by chance

For example, genetic models can predict the probability that offspring inherit particular combinations of alleles.

These probabilities describe possible outcomes across many cases, not what must happen to one individual.

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Probability in Genetics

Suppose two parents carry alleles that produce the following theoretical probabilities for a particular simplified genetic cross:

25% AA

50% Aa

25% aa

This does not mean that four children must include:

• exactly one AA
• exactly two Aa
• exactly one aa

Each birth represents another outcome.

The probabilities describe the model, not a required family pattern.

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Probability in Technology

Probability is important in modern technology.

It is used in:

• artificial intelligence
• computer security
• spam filtering
• recommendation systems
• speech recognition
• weather models
• robotics
• quality control
• communications

For example, a computer system may estimate probabilities for several possible interpretations of a spoken word and choose the most likely interpretation based on its model.

---

Probability in Decision-Making

Imagine two choices.

Choice A

90% probability of gaining $10.

Choice B

10% probability of gaining $100.

The probabilities alone do not automatically tell someone which choice they should prefer.

People may consider:

• possible reward
• possible loss
• importance of the outcome
• personal circumstances
• tolerance for risk

Probability provides mathematical information.

Decision-making combines that information with consequences, goals, and values.

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Expected Value

One tool for comparing uncertain outcomes is expected value.

A simple expected value calculation multiplies each possible outcome by its probability and combines the results.

Suppose a game has:

50% chance of winning $10
50% chance of winning $0

Expected value:

(0.5 × $10) + (0.5 × $0)

= $5

This does not mean you will receive $5 each time.

You will receive either:

$10 or $0

The $5 represents the average outcome expected across many repetitions of the same probability model.

---

Worked Example: Expected Value

Suppose a game has:

25% chance of receiving $20
75% chance of receiving $0

Expected value:

(0.25 × $20) + (0.75 × $0)

= $5

Again, you never actually receive $5 from one play.

Expected value describes the long-run mathematical average.

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Probability and Personal Decisions

People regularly make informal probability judgments.

For example:

"Traffic is usually heavy at this time."

"The clouds look like rain."

"This bus is normally reliable."

"I usually finish this journey in 20 minutes."

These judgments are often based on past experience.

They are forms of informal probability reasoning.

Better decisions can often be made when we combine experience with reliable data.

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Probability Does Not Tell the Whole Story

Suppose two events each have:

P(event) = 0.01

or:

1%

Event A:

A 1% chance of losing a pencil.

Event B:

A 1% chance of a serious equipment failure.

The probabilities are identical.

The consequences are not.

This demonstrates why real-world decisions often require more than just calculating probability.

We must consider:

Probability + Consequence + Context

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Large Numbers Can Make Small Probabilities Important

A small probability can become important when an event is repeated many times.

Suppose a manufactured component has:

0.1% probability of being defective

That seems very small.

But if a factory produces:

1,000,000 components

expected defects:

1,000,000 × 0.001

= 1,000

A small probability applied to a very large number of events can produce a substantial expected number of occurrences.

---

Worked Example: Flight Delays

Suppose historical data show that:

15% of similar flights are delayed

A traveller has an important connection.

The probability of delay is:

0.15

The probability of no delay is:

1 − 0.15

= 0.85

= 85%

Although arriving without delay is more likely, the traveller might still allow extra time because missing the connection could have significant consequences.

This illustrates how probability and consequences interact in decision-making.

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Comparing Risks Fairly

When comparing probabilities, use the same form.

Suppose:

Risk A = 1 in 20

Risk B = 8%

Convert:

1/20 = 0.05 = 5%

Now compare:

Risk A = 5%

Risk B = 8%

Risk B has the larger probability.

Converting probabilities to the same format makes comparisons easier.

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Probability Formats in the Media

Probability information can appear as:

• fractions
• decimals
• percentages
• ratios
• "1 in n" statements

For example:

1/4

0.25

25%

1 in 4

all represent the same probability.

Being able to move between these forms makes it easier to interpret real-world information.

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Be Careful with Headlines

A headline might say:

"Risk doubles!"

That sounds dramatic.

But always ask:

From what probability to what probability?

An increase from:

1% to 2%

is a doubling.

An increase from:

30% to 60%

is also a doubling.

The changes have very different practical meanings.

Good probability thinking requires examining the actual numbers rather than relying only on descriptive language.

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Probability and Sample Size

Suppose a website claims:

"90% of people prefer Product A!"

Before accepting the claim, ask:

How many people were surveyed?

If:

9 out of 10 people

preferred Product A, that is 90%.

If:

9,000 out of 10,000 people

preferred Product A, that is also 90%.

The percentage is the same, but the amount of evidence is very different.

Larger samples often provide more stable estimates, provided the sample was selected appropriately.

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Bias Matters

A large sample can still give misleading information if it is biased.

Suppose a school wants to know whether students enjoy basketball.

If researchers survey only members of the basketball team, the sample is unlikely to represent the whole school.

Probability and statistics depend on:

• good data
• appropriate sampling
• careful measurement
• reasonable assumptions

A precise-looking percentage is not automatically trustworthy.

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Making Better Probability Decisions

When using probability in real life, ask:

What is the probability?

Where did the number come from?

Is it theoretical or based on observations?

How much data were collected?

Is the sample representative?

What assumptions were made?

What are the consequences if the event occurs?

What are the consequences if it does not occur?

How certain is the estimate?

These questions help turn probability from a calculation into a useful decision-making tool.

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Common Mistakes

Thinking likely means certain

A 90% probability still allows the event not to occur.

Thinking unlikely means impossible

A 1% event can still happen.

Assuming probability predicts individual outcomes

Probability describes likelihood, not exactly what will happen next.

Misinterpreting weather probabilities

A percentage chance of rain does not simply mean it will rain for that percentage of the day.

Ignoring consequences

A low-probability event may still deserve attention if its consequences are severe.

Looking only at relative changes

"Risk doubled" is difficult to interpret without knowing the original probability.

Ignoring sample size

A percentage based on very little data may be unstable.

Ignoring bias

A large dataset can still be misleading if the method of collecting it is biased.

Treating expected values as guaranteed outcomes

An expected value describes a long-run average, not necessarily an outcome that will actually occur.

Assuming past performance guarantees future performance

Historical data can help estimate probabilities, but conditions may change.

Confusing expected frequency with actual frequency

Expected frequency is a prediction based on probability.

Actual frequency is what really occurred.

Ignoring the source of a probability

Probabilities based on strong evidence may be more useful than unsupported estimates.

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Key Terms

Probability — A measure of how likely an event is to occur.

Likelihood — The chance that an event will happen.

Uncertainty — A situation in which an outcome cannot be known with complete certainty.

Risk — The possibility of an unwanted event, often considered together with its consequences.

Consequence — The result or impact of an event occurring.

Risk assessment — The process of considering the likelihood and consequences of possible events.

Theoretical probability — Probability calculated using a mathematical model.

Experimental probability — Probability estimated from observed results.

Empirical probability — Another term for probability based on observed data.

Relative frequency — The proportion of observations in which an event occurs.

Expected frequency — The predicted number of times an event should occur based on probability.

Expected value — A probability-weighted long-run average of possible numerical outcomes.

Forecast — A prediction about future conditions based on available information and models.

Probability of precipitation — A weather probability describing the chance of measurable precipitation for the defined forecast period and location or area.

Sample — A group selected from a larger population for study.

Sample size — The number of observations or individuals in a sample.

Bias — A systematic influence that can make results unrepresentative or favour particular outcomes.

Random variation — Natural differences that occur because of chance.

Absolute risk — The actual probability of an event occurring.

Relative change — A comparison showing how much a value has changed relative to its original value.

Risk factor — A characteristic associated with a change in the probability of an outcome.

Probability model — A mathematical description of possible outcomes and their probabilities.

Historical data — Information collected from events that have already occurred.

Prediction — A statement or estimate about a future outcome.

Long-run average — The average result expected when a random process is repeated many times.

Decision-making — Choosing between possible actions using available information, goals, consequences, and other relevant factors.

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Key Takeaways

• Probability is used throughout everyday life to describe uncertainty.
• Weather forecasts use probability to describe the likelihood of future weather events.
• A high probability does not guarantee an event will occur.
• A low probability does not mean an event is impossible.
• Sports statistics can provide experimental estimates of future performance.
• Historical data can help estimate probabilities, but future conditions may differ.
• Probability is important when assessing risk.
• Risk involves both likelihood and consequence.
• A low-probability event can still be important if its consequences are serious.
• Insurance companies use probability and large datasets to estimate risks.
• Transportation systems use probability to study delays, reliability, accidents, and demand.
• Manufacturers use probability to estimate defect rates and monitor quality.
• Scientists and medical researchers use probability to analyze uncertain outcomes.
• Natural-hazard probabilities describe likelihood, not fixed schedules.
• A "100-year" event does not mean the event can happen only once every 100 years.
• Expected frequency predicts approximately how often an event should occur.
• Expected frequency does not guarantee the actual frequency.
• Expected value describes a long-run probability-weighted average.
• Small probabilities can become important when events are repeated many times.
• Probabilities can be written as fractions, decimals, percentages, or "1 in n" statements.
• Converting probabilities to the same form makes comparisons easier.
• Relative changes can sound dramatic, so actual probabilities should also be examined.
• Sample size matters when interpreting probabilities estimated from data.
• Bias can make probability estimates misleading even when a sample is large.
• Probability provides information for decision-making, but it does not make decisions for us.
• Good decisions may also consider consequences, costs, benefits, uncertainty, and context.

When you encounter probability in everyday life, ask:

What exactly is the event?

How likely is it?

Where did the probability come from?

How much evidence supports it?

What assumptions were made?

What happens if the event occurs?

What happens if it does not occur?

How should this information affect the decision?

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Check Your Understanding

1. Give three examples of probability being used in everyday life.

2. A weather forecast gives an 80% chance of rain. Explain what this tells you.

3. Does an 80% chance of rain guarantee rain? Explain.

4. Explain why a 20% probability does not mean an event is impossible.

5. A basketball player makes 84 of 120 free throws. Calculate the experimental probability of making a free throw.

6. Express your answer to Question 5 as a percentage.

7. Use the player's previous results to predict approximately how many successful shots might occur in another 200 attempts under similar conditions.

8. Explain why your answer to Question 7 is not guaranteed.

9. A bus arrives on time for 450 of 500 recorded journeys. Calculate the experimental probability of an on-time arrival.

10. Calculate the probability of a late arrival based on these data.

11. A factory produces 4,000 products and finds 60 defective items. Calculate the experimental probability of a defect.

12. If conditions remain similar, estimate the number of defective products expected among 20,000 products.

13. Explain the difference between probability and consequence when assessing risk.

14. Give an example of a low-probability event that could still have serious consequences.

15. A report states that a risk increased from 1% to 2%.

a. By how many percentage points did the risk increase?
b. By what factor did the risk increase?
c. Explain why both pieces of information are useful.

16. Convert each probability to a percentage:

a. 1/4
b. 0.08
c. 3/5
d. 1/100

17. Convert each to a decimal:

a. 35%
b. 2%
c. 75%
d. 0.5%

18. Risk A is 1 in 20. Risk B is 8%. Which has the greater probability? Show your calculation.

19. Explain why a "100-year flood" does not mean that such a flood occurs exactly once every 100 years.

20. A game gives a 25% chance of winning $20 and a 75% chance of winning nothing. Calculate the expected value.

21. Explain why the expected value in Question 20 does not mean a player receives that amount every time.

22. A website reports that 90% of surveyed people prefer Product A. Give three questions you should ask before deciding whether this is strong evidence.

23. Explain why sample size matters when interpreting experimental probability.

24. Explain why a biased sample can be misleading even when the sample is large.

25. A component has a 0.2% defect probability. Express this as:

a. a decimal
b. a fraction out of 1,000

26. A factory produces 500,000 of the components from Question 25. Estimate the expected number of defective components.

27. Give two reasons why a sports player's historical success rate might not perfectly predict future performance.

28. Explain why probability should inform decisions rather than be treated as a guarantee.

29. A traveller has two possible buses:

Bus A historically arrives on time 92% of the time.

Bus B historically arrives on time 78% of the time.

List at least three additional pieces of information that could matter when deciding which bus to take.

30. Challenge: You are organizing an outdoor school event. The forecast gives a 65% probability of rain. Moving the event indoors would require extra preparation, but heavy rain could disrupt the outdoor event.

Explain how you would use probability to think about the decision. Your answer should discuss:

• what the 65% probability means
• why rain is not guaranteed
• the possible consequences of rain
• the consequences of moving indoors
• what additional information would be useful
• why probability can inform the decision without determining it automatically.