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Lesson overview

Statistics and data — probability

Probability measures how likely an event is, on a scale from 0 to 1. If all outcomes are equally likely, $P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of possible outcomes}}$ . The complement of an event A is 'not A', and $P(\text{not A}) = 1 - P(A)$ . Expected frequency = probability × number of trials. For example, if the probability of rolling a 6 on a fair die is $\frac{1}{6}$ , then in 60 rolls you'd expect to see about 10 sixes.

Calculating theoretical probability

A fair die is rolled once. What is the probability of rolling a number greater than 4?

The probability is the ratio of favourable outcomes to total outcomes. Always simplify the fraction.

Using the complement rule P(not A) = 1 - P(A)

A spinner has a 0.3 probability of landing on red. What is the probability it does NOT land on red?

RuleThe probability of an event and its complement always sum to 1

If something has a 30% chance of happening, it has a 70% chance of NOT happening.

Expected frequency in trials

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

AnswerYou expect about 100 heads

Expected frequency = probability × number of trials. Remember: this is what you expect on average, not what you'll always get.

Worked example — fair game or unfair game?

A game uses two fair dice. You win £1 if the sum is 7, but lose £0.50 if it is not. Is this a fair game?

ConclusionUnfair — the player expects to lose £0.25 per game on average

List all ways to get a sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — that's 6 ways.
Total possible outcomes with 2 dice: 6 × 6 = 36.
Probability of getting 7: P(7) = $\frac{6}{36}$ = $\frac{1}{6}$ .
Probability of NOT getting 7: P(not 7) = 1 - $\frac{1}{6}$ = $\frac{5}{6}$ .
Calculate expected value: ( $\frac{1}{6}$ ) × £1 + ( $\frac{5}{6}$ ) × (−£0.50).
Expected value = $\frac{1}{6}$ - 2. $\frac{50}{6}$ = -1. $\frac{50}{6}$ ≈ -£0.25.
Since the expected value is negative, the game is unfair — players lose money on average.

Try it

Calculate probabilities, use the complement rule, and determine whether games are fair or unfair.

Question 1

A bag contains 3 red, 4 blue and 5 green marbles. What is the probability of drawing a red marble?

💡 Total marbles = 3 + 4 + 5 = 12. Red marbles = 3. Probability = $\frac{3}{12}$ = $\frac{1}{4}$ .

Question 2

If P(A) = 0.4, what is P(not A)?

💡 P(not A) = 1 - P(A) = 1 - 0.4 = 0.6.

Question 3

A fair coin is flipped 500 times. How many tails do you expect?

💡 P(tails) = 0.5. Expected frequency = 0.5 × 500 = 250.

Question 4

A fair spinner has 8 equal sections, 2 of which are blue. What is P(blue)?

💡 P(blue) = $\frac{2}{8}$ = $\frac{1}{4}$ .

Question 5

A die is rolled 300 times. How many times would you expect to roll a number less than 3?

💡 Numbers less than 3: 1, 2 (2 outcomes). P(less than 3) = $\frac{2}{6}$ = $\frac{1}{3}$ . Expected = ( $\frac{1}{3}$ ) × 300 = 100.

Common mistakes

Watch for these when working through the lesson.

Forgetting to simplify probability fractions: $\frac{6}{36}$ should be simplified to $\frac{1}{6}$ .
Confusing P(A) with P(not A): remember that P(A) + P(not A) = 1.
Treating expected frequency as a guaranteed result: expected frequency is an average over many trials, not what happens in one trial.

Probabilities of single events

Lesson overview

Worked example — fair game or unfair game?

Try it

Common mistakes

Related topics

Independent practice will plug in here