GCSE Maths revision · Probability and statistics

Probability and expected outcomes

Foundation and Higher

The idea

Probability runs from $0$ (impossible) to $1$ (certain). If an event has probability $p$ , in $n$ trials you expect it about $np$ times.

Probability $= \frac{\text{number of ways it can happen}}{\text{total number of outcomes}}$ when outcomes are equally likely.
Expected number $=$ probability $\times$ number of trials.
In a frequency tree, the branches from one box add up to that box.

Watch out Expected does not mean guaranteed: it is a long-run average.

See it worked

Worked exampleNon-calculator2 marks

A counter is taken from a bag, its colour is recorded, and it is replaced. This is repeated 40 times. Red is recorded 17 times and blue 14 times. Every other result is yellow. How many times is yellow recorded?

$40-17-14$
The frequencies of all outcomes add to the total number of trials.
Yellow frequency = 40 $-$ 17 $-$ 14 = 9.

Answer $9$

Your turn

Question 1Calculator allowed2 marks

A fair eight-sided die has faces numbered 1 to 8. It is rolled 360 times. How many times would you expect a number greater than 5?

Question 2Non-calculator4 marks

36 pupils are invited to a club. 27 attend, including 15 boys. Everyone attending is recorded as either a boy or a girl.

(a) Find the number of girls attending.

(b) A pupil from the invited group is selected at random. Find the probability they attended.

Question 3Non-calculator4 marks

A biased dice is thrown. The probabilities of 1, 2, 3, 4 and 5 are each 0.15. At a fair, each throw costs 60p. A player who throws a 6 wins £2. Otherwise the player wins nothing.

The dice is thrown 300 times. Work out an estimate of the organiser's profit.

In past papers

140 official questions in 104 papers since 2017. The latest:

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