Expected value — Expectation — Core Maths Compendium

See it first

Watch it happen

Scroll to the graphic and it plays by itself — press Replay to watch it again.

Build it up, step by step

Expected value — the key ideas

Click each step below to reveal it — work through them in order the first time round.

Step 1 · Expected value

1 of 4▾

The expected value of a random variable $X$ is its long-run average: $E(X) = \sum x\,P(X = x)$. Multiply each value by its probability and add them all up. It does not have to be a value $X$ can actually take.

Step 2 · Expected frequency

2 of 4▾

If an event has probability $p$ and there are $n$ trials, the expected frequency is $n \times p$. For example, if the chance of rain on any day is 0.15, you expect about $40 \times 0.15 = 6$ rainy days in 40 days.

Step 3 · Fair games

3 of 4▾

For a game, use the net winnings (prize − stake). A game is fair if the expected net winnings are 0. If they are positive the player expects to gain; if negative, the organiser expects to profit by that amount each play.

Step 4 · Using expected value to decide

4 of 4▾

To compare options, calculate each expected value and choose the better one — but comment on risk: an expected value is a long-run average and says nothing about the spread of outcomes. A one-off decision with a small chance of a huge loss may not be sensible even if the expected value is positive.

Worked example

A game costs £2 to play. A fair dice is rolled: a 6 wins £10, otherwise nothing. Find the expected net winnings.

Net winnings: +£8 with probability $\frac{1}{6}$; −£2 with probability $\frac{5}{6}$.

$E(X) = 8 \times \frac{1}{6} - 2 \times \frac{5}{6} = \frac{8}{6} - \frac{10}{6} = -\frac{1}{3}$. The player loses about 33p per game; the game is not fair.

Test yourself

Past-paper style question

At a fun-fair a game costs £2 to play. The prize is £0 with probability 0.6, £2 with probability 0.3 and £5 with probability 0.1.

(a) Find the expected prize.
(b) Find the organiser's expected profit per game.
(c) 200 people play. Find the expected total profit.
(d) Is the game fair? Give a reason. [5 marks]

Show the answer

▾

(a) $E(\text{prize}) = 0(0.6) + 2(0.3) + 5(0.1) = 0 + 0.6 + 0.5 = $ £1.10. (2 marks)

(b) Profit = stake − expected prize = 2 − 1.10 = £0.90. (1 mark)

(c) $200 \times 0.90 = $ £180. (1 mark)

(d) Not fair: the expected net winnings for a player are −£0.90, not 0. (1 mark)

Practice

Expected value worksheet

Five short questions. Work through them, then reveal the mark scheme to check.

$X$ takes the values 1, 2, 3 with probabilities 0.2, 0.5, 0.3. Find $E(X)$.
The probability of rain on any day is 0.15. How many rainy days are expected in 40 days?
A coin is tossed. Heads wins £5, tails wins nothing. It costs £3 to play. Find the expected net winnings.
A game pays £12 with probability $\frac{1}{4}$ and nothing otherwise. What stake makes the game fair?
$X$ takes the values 0, 10, 20 with probabilities 0.5, 0.3, 0.2. Find $E(X)$.

Mark scheme

▾

$1(0.2) + 2(0.5) + 3(0.3) = 0.2 + 1 + 0.9 = $ 2.1.
$40 \times 0.15 = $ 6 days.
Net: +£2 (heads) or −£3 (tails), each with probability 0.5: $E = 1 - 1.5 = $ −£0.50.
Expected prize = $12 \times \frac{1}{4} = 3$. A fair stake is £3.
$0(0.5) + 10(0.3) + 20(0.2) = 0 + 3 + 4 = $ 7.

← Back to Expectation