What Is Probability?

Probability is a measure of how likely an event is to happen.

We use probability whenever there is uncertainty about an outcome.

Examples include:

• whether a coin lands heads or tails
• which number appears when a die is rolled
• which color is selected from a bag
• whether it will rain
• whether a machine produces a defective item
• how often a particular result may occur in repeated trials

Probability does not usually tell us exactly what will happen next. Instead, it describes how likely different outcomes are.

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The Probability Scale

Probability ranges from:

0 to 1

or:

0% to 100%

Important points include:

0 = 0% = impossible

1/4 = 0.25 = 25% = unlikely

1/2 = 0.5 = 50% = even chance

3/4 = 0.75 = 75% = likely

1 = 100% = certain

A probability can never be less than 0 or greater than 1.

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Probability as a Fraction

When outcomes are equally likely:

Probability = Number of favorable outcomes / Total number of possible outcomes

We often write this as:

P(event) = favorable outcomes / total outcomes

For example, a fair six-sided die has six possible outcomes:

1, 2, 3, 4, 5, 6

The probability of rolling a 4 is:

P(4) = 1/6

There is one favorable outcome out of six possible outcomes.

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Probability as a Decimal

A fraction can be converted to a decimal by dividing:

numerator ÷ denominator

For example:

1/4 = 1 ÷ 4 = 0.25

Therefore:

P = 0.25

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Probability as a Percentage

To convert a decimal probability into a percentage:

Decimal × 100%

For example:

0.25 × 100% = 25%

Therefore:

1/4 = 0.25 = 25%

All three values describe exactly the same probability.

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Common Probability Equivalents

Recognizing these common equivalents makes probability comparisons much easier.

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Simple Probability

Suppose a bag contains:

3 red counters

2 blue counters

5 counters altogether

What is the probability of selecting red?

Favorable outcomes:

3

Total outcomes:

5

Therefore:

P(red) = 3/5

Convert to decimal:

3 ÷ 5 = 0.6

Convert to percentage:

0.6 × 100% = 60%

Therefore:

P(red) = 3/5 = 0.6 = 60%

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Probability with a Fair Coin

A fair coin has two possible outcomes:

Heads

Tails

Therefore:

P(heads) = 1/2

and:

P(tails) = 1/2

Convert:

1/2 = 0.5 = 50%

So each outcome has a 50% probability on a single fair toss.

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Probability with a Fair Die

A standard fair die has six possible outcomes:

1, 2, 3, 4, 5, 6

Each individual number has probability:

1/6

What is the probability of rolling an even number?

Even numbers:

2, 4, 6

There are:

3 favorable outcomes

Therefore:

P(even) = 3/6

Simplify:

P(even) = 1/2

So:

P(even) = 0.5 = 50%

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Another Die Example

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

Possible favorable outcomes:

5, 6

Therefore:

P(number > 4) = 2/6

Simplify:

1/3

Decimal:

1 ÷ 3 ≈ 0.333

Percentage:

≈ 33.3%

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Probability with a Spinner

Suppose a spinner has:

8 equal sections

with:

3 red

2 blue

2 green

1 yellow

Probability of red:

3/8

Decimal:

0.375

Percentage:

37.5%

Probability of yellow:

1/8

Decimal:

0.125

Percentage:

12.5%

Because red occupies more equal sections than yellow, red is more likely to occur.

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Equally Likely Outcomes

The simple formula:

P(event) = favorable outcomes / total outcomes

works directly when the individual outcomes are equally likely.

For example, with a fair die, each face has the same probability.

But suppose a spinner contains sections of different sizes.

A large section may be more likely than a small section.

In that situation, simply counting the number of sections may not correctly determine probability.

The size or probability of each outcome must also be considered.

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Favorable Outcomes

A favorable outcome is an outcome that satisfies the event we are interested in.

Suppose a die is rolled.

Event:

Roll a number less than 5

Favorable outcomes:

1, 2, 3, 4

Therefore:

P(number < 5) = 4/6

Simplify:

2/3

This does not mean that 1, 2, 3, and 4 are "good." In probability, favorable simply means that the outcome matches the event being investigated.

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Impossible Events

An impossible event has probability:

0

For example, rolling a 9 on a standard six-sided die is impossible.

Therefore:

P(9) = 0

or:

0%

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Certain Events

A certain event has probability:

1

For example, rolling a number less than 7 on a standard six-sided die is certain.

Possible outcomes:

1, 2, 3, 4, 5, 6

All six satisfy the condition.

Therefore:

P(number < 7) = 6/6 = 1 = 100%

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Likelihood Language

Probability can also be described using words.

Impossible: probability = 0

Unlikely: probability closer to 0 than 0.5

Even chance: probability = 0.5

Likely: probability closer to 1 than 0.5

Certain: probability = 1

These words describe general levels of likelihood rather than always specifying an exact probability.

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Comparing Probabilities

Suppose:

Event A:

P(A) = 1/4

Event B:

P(B) = 0.40

Event C:

P(C) = 35%

Convert them to the same form.

Event A:

1/4 = 25%

Event B:

0.40 = 40%

Event C:

35%

Now the probabilities are:

A = 25%

B = 40%

C = 35%

Putting probabilities in the same form makes them much easier to compare.

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Comparing Fractions

Suppose:

P(A) = 2/5

and:

P(B) = 3/8

Convert to decimals:

2/5 = 0.40

3/8 = 0.375

Therefore:

0.40 > 0.375

So event A has the greater probability.

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Complementary Events

The complement of an event is the event not happening.

For any event A:

P(A) + P(not A) = 1

or:

100%

Suppose:

P(rain) = 30%

Then:

P(no rain) = 100% − 30%

= 70%

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Complement Example with a Die

What is the probability of not rolling a 6?

Probability of rolling 6:

1/6

Therefore:

P(not 6) = 1 − 1/6

= 5/6

or approximately:

83.3%

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Theoretical Probability

Theoretical probability is based on a mathematical model of possible outcomes.

For a fair coin:

P(heads) = 1/2

For a fair six-sided die:

P(6) = 1/6

These probabilities can be calculated before conducting an experiment.

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Experimental Probability

Experimental probability is based on actual observations or trials.

The equation is:

Experimental Probability = Number of Times Event Occurs / Total Number of Trials

Suppose a coin is tossed 50 times.

Heads occurs:

27 times

Experimental probability:

27/50

= 0.54

= 54%

The theoretical probability is 50%, but an experiment does not have to produce exactly 50%.

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Why Experimental Results Vary

Random processes naturally produce variation.

A fair coin tossed 10 times might produce:

7 heads and 3 tails

That does not necessarily mean:

P(heads) = 70%

The theoretical probability remains:

50%

for a fair coin.

With a larger number of trials, experimental proportions often become closer to the theoretical probability.

This idea is connected to the law of large numbers.

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Relative Frequency

Experimental probability is also called relative frequency.

Suppose a basketball player takes:

80 free throws

and makes:

60

Relative frequency of success:

60/80

= 3/4

= 0.75

= 75%

This information can be used to describe past performance and may help estimate future outcomes, while recognizing that future results are uncertain.

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Using Probability to Make Predictions

Probability can help estimate how many times an event may occur over many trials.

Suppose:

P(red) = 0.30

and a random selection experiment is performed:

200 times

Expected number of red outcomes:

0.30 × 200 = 60

Therefore, we might predict:

about 60 red outcomes

This is a prediction, not a guarantee.

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Expected Frequency

The expected number of times an event occurs can be calculated using:

Expected Frequency = Probability × Number of Trials

For example:

Probability of success:

0.4

Number of trials:

150

Expected frequency:

0.4 × 150 = 60

We would expect approximately:

60 successes

over many similar trials.

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Prediction with Fractions

Suppose:

P(blue) = 3/5

and there will be:

100 selections

Expected blue outcomes:

3/5 × 100

= 60

Prediction:

about 60 blue outcomes

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Prediction with Percentages

Suppose a machine historically produces:

2% defective items

If:

5,000 items

are produced under similar conditions, a simple prediction based on that rate is:

2% of 5,000

0.02 × 5000 = 100

So the expected number is:

about 100 defective items

Actual results could be higher or lower.

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Probability in Weather Information

Weather forecasts often use probabilities.

Suppose a forecast reports:

70% chance of rain

This describes a probability associated with the forecast event and its specified place and time period.

It does not mean that it will rain for exactly 70% of the day.

It also does not guarantee that rain will occur.

A probability describes uncertainty, not certainty.

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Probability in Sports

Sports statistics often use past frequencies.

Suppose a player successfully completes:

42 of 60 attempts

Success rate:

42/60

Simplify:

7/10

Decimal:

0.70

Percentage:

70%

If the player makes another 100 attempts under broadly similar conditions, 70 successes might be used as a simple prediction based on the historical rate.

But actual performance can change.

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Probability in Games

Games frequently involve probability.

Examples include:

• dice
• cards
• spinners
• coins
• random number generators

Suppose a game awards a prize if a player rolls a 6.

Probability of winning:

1/6

Probability of not winning:

5/6

Knowing these probabilities can help us understand how the game is structured.

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Probability with Cards

A standard deck contains:

52 cards

There are:

4 suits

with:

13 cards in each suit

Probability of drawing a heart:

13/52

Simplify:

1/4

Therefore:

P = 0.25 = 25%

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Another Card Example

There are four aces in a standard 52-card deck.

Therefore:

P(ace) = 4/52

Simplify:

1/13

Decimal:

≈ 0.077

Percentage:

≈ 7.7%

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Probability from Tables

Suppose a survey records preferred transportation:

Total:

24 + 36 + 20 + 20 = 100

If one student is randomly selected:

P(bus) = 36/100 = 0.36 = 36%

P(bicycle) = 20/100 = 0.20 = 20%

Tables can therefore be used to calculate probabilities from data.

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Probability from Graphs

Graphs can also provide information used to estimate probabilities.

Suppose a graph shows:

Red = 40 outcomes

Blue = 30 outcomes

Green = 20 outcomes

Yellow = 10 outcomes

Total:

100 outcomes

Therefore:

P(red) = 40/100 = 40%

P(blue) = 30/100 = 30%

P(green) = 20/100 = 20%

P(yellow) = 10/100 = 10%

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Interpreting Probability Data Carefully

Suppose:

20 people are surveyed.

16 choose option A.

Experimental proportion:

16/20 = 80%

It would be correct to say:

80% of the surveyed group chose A.

It would require additional assumptions to claim that exactly 80% of a much larger population would choose A.

The size and selection of the sample matter when probability and statistics are used to make predictions.

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Probability Does Not Guarantee Individual Outcomes

Suppose:

P(win) = 90%

This means winning is very likely under the stated model.

It does not mean winning is certain.

A 10% probability of not winning still exists.

Similarly:

P(event) = 1%

does not mean the event is impossible.

Low probability and impossibility are different ideas.

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Random Does Not Mean "Equal"

A process can be random without every outcome having the same probability.

For example, imagine a spinner where:

70% of the area is blue

and:

30% is red.

The result may still be random, but:

P(blue) = 0.70

and:

P(red) = 0.30

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Independent Repeated Events

Suppose a fair coin has landed heads five times in a row.

If each toss is independent, the probability of heads on the next toss is still:

1/2

The coin does not "owe" us a tail.

Previous independent results do not change the probability of the next toss.

This common misunderstanding is sometimes called the gambler's fallacy.

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Probability and Risk

Probability is often used to describe risk.

For example, suppose two hypothetical events have probabilities:

Event A:

2%

Event B:

20%

Event B is ten times as likely under the stated model:

20% ÷ 2% = 10

However, probability is only one part of evaluating risk. The consequences of an event may also matter.

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Worked Example 1: Colored Counters

A bag contains:

4 red

5 blue

1 yellow

Total:

10

Probability of blue:

5/10 = 1/2

Decimal:

0.5

Percentage:

50%

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Worked Example 2: Die

What is the probability of rolling a number greater than 2?

Favorable outcomes:

3, 4, 5, 6

Therefore:

P(>2) = 4/6

Simplify:

2/3

Decimal:

≈ 0.667

Percentage:

≈ 66.7%

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Worked Example 3: Spinner

A spinner has 20 equal sections.

Eight are green.

Probability:

8/20

Simplify:

2/5

Decimal:

0.4

Percentage:

40%

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Worked Example 4: Compare Probabilities

Event A:

3/4

Event B:

0.70

Event C:

72%

Convert:

A = 75%

B = 70%

C = 72%

So their probabilities are:

75%, 70%, and 72%

Converting to a common representation makes the comparison clear.

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Worked Example 5: Complement

Probability of success:

0.65

Probability of failure:

1 − 0.65 = 0.35

Therefore:

P(failure) = 35%

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Worked Example 6: Experimental Probability

A spinner is used:

200 times

Blue occurs:

46 times

Experimental probability:

46/200

= 0.23

= 23%

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Worked Example 7: Prediction

Suppose:

P(success) = 35%

There will be:

400 trials

Expected successes:

0.35 × 400 = 140

Prediction:

about 140 successes

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Worked Example 8: Using Historical Data

A sports player succeeds:

72 times in 90 attempts

Experimental probability:

72/90

= 0.8

= 80%

If the player makes another 50 attempts under similar conditions:

Expected successes:

0.80 × 50 = 40

A simple prediction is:

about 40 successes

This remains an estimate rather than a guarantee.

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Worked Example 9: Probability from a Table

A survey gives:

Total:

100

If one response is randomly selected:

P(A) = 45%

P(B) = 30%

P(C) = 15%

P(D) = 10%

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Worked Example 10: Multi-Step Probability

A bag contains:

6 red

3 blue

1 green

Total:

10

Probability of selecting something that is not red:

Blue or green outcomes:

3 + 1 = 4

Therefore:

P(not red) = 4/10

= 2/5

= 0.4

= 40%

We could also use the complement:

P(red) = 6/10 = 60%

Therefore:

P(not red) = 100% − 60% = 40%

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Making Predictions from Probability

Probability-based predictions are most useful over many trials.

Suppose:

P(A) = 0.25

For 20 trials:

Expected frequency:

0.25 × 20 = 5

For 100 trials:

0.25 × 100 = 25

For 1,000 trials:

0.25 × 1000 = 250

The actual results may differ from these numbers, especially for smaller numbers of trials.

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A Reliable Probability Strategy

Step 1: Identify the event.

What exactly are you trying to find?

Step 2: Identify all possible outcomes.

Step 3: Determine whether the outcomes are equally likely.

Step 4: Count the favorable outcomes.

Step 5: Calculate:

P(event) = favorable outcomes / total outcomes

Step 6: Simplify the fraction if possible.

Step 7: Convert to a decimal or percentage if required.

Step 8: Check that the probability is between 0 and 1.

Step 9: Interpret what the probability means in context.

Step 10: If making a prediction, remember that expected results are not guaranteed results.

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Common Mistakes

Mistake 1: Writing a probability greater than 1

For example:

P(A) = 1.4

cannot be a valid probability.

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Mistake 2: Forgetting the total number of outcomes

If a bag contains 3 red and 7 blue counters:

P(red) = 3/10

not:

3/7

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Mistake 3: Confusing percentages and decimals

25% = 0.25

not:

25

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Mistake 4: Assuming all outcomes are equally likely

This is only valid when the model supports that assumption.

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Mistake 5: Treating a prediction as a guarantee

A predicted 60 successes does not mean exactly 60 must occur.

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Mistake 6: Treating unlikely as impossible

A probability of 1% is small, but it is not zero.

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Mistake 7: Treating likely as certain

A probability of 90% is high, but it is not 100%.

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Mistake 8: Assuming short experiments must match theoretical probability

Ten fair coin tosses do not have to produce exactly five heads.

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Mistake 9: Assuming previous independent results control the next result

Five heads in a row do not make tails "due" on the next independent toss.

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Did You Know?

Probability connects many areas of mathematics and real-world decision-making.

Probability is used in:

• weather forecasting
• science
• genetics
• medicine and public health research
• engineering
• manufacturing
• sports statistics
• insurance
• economics
• quality control
• computer science
• games

In each case, probability helps describe uncertainty.

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Key Terms

• Probability: Measure of how likely an event is.
• Outcome: One possible result.
• Event: One outcome or a collection of outcomes being investigated.
• Favorable outcome: Outcome that satisfies the event being studied.
• Equally likely: Outcomes having the same probability.
• Impossible event: Event with probability 0.
• Certain event: Event with probability 1.
• Likelihood: How probable an event is.
• Complement: Event consisting of an event not occurring.
• Theoretical probability: Probability calculated from a mathematical model.
• Experimental probability: Probability estimated from observed results.
• Relative frequency: Proportion of trials in which an event occurs.
• Trial: One performance of a probability experiment.
• Expected frequency: Predicted number of occurrences based on probability.
• Random: Involving uncertainty in individual outcomes.
• Independent events: Events where the outcome of one does not affect the probability of another.
• Prediction: Estimate of future results based on available information.

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Key Equations and Rules

Simple probability for equally likely outcomes:

P(event) = Favorable Outcomes / Total Possible Outcomes

Experimental probability:

P(event) = Number of Times Event Occurs / Total Number of Trials

Complement rule:

P(A) + P(not A) = 1

Therefore:

P(not A) = 1 − P(A)

Expected frequency:

Expected Frequency = Probability × Number of Trials

Fraction to decimal:

Numerator ÷ Denominator

Decimal to percentage:

Decimal × 100%

Percentage to decimal:

Percentage ÷ 100

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Key Takeaways

• Probability measures how likely an event is to occur.
• Probabilities range from 0 to 1, or from 0% to 100%.
• A probability of 0 represents an impossible event.
• A probability of 1 represents a certain event.
• An even chance has probability 1/2 = 0.5 = 50%.
• Probabilities can be written as fractions, decimals, or percentages.
• Different forms can represent exactly the same probability.
• For equally likely outcomes, probability can be calculated using favorable outcomes divided by total possible outcomes.
• Probabilities are easier to compare when converted to the same form.
• Complementary probabilities add to 1, or 100%.
• Theoretical probability comes from a mathematical model.
• Experimental probability comes from observed data.
• Experimental results may differ from theoretical probabilities.
• Larger numbers of trials often produce experimental proportions closer to theoretical probabilities.
• Expected frequency can be calculated by multiplying probability by the number of trials.
• Probability can be used to make predictions, but predictions are not guarantees.
• Random does not necessarily mean that all outcomes are equally likely.
• Past independent outcomes do not make a particular future outcome "due."
• Real-world probability information should be interpreted in context, especially when it comes from samples or historical data.
• Probability is useful because it gives us a mathematical way to describe and reason about uncertainty.