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.