IB Maths AI HL
Probability Distributions
Paper 1 & 2
~6 min read
Expected Values E(X)
The expected value E(X) is the mean of a random variable — the long-run average outcome if you repeated the experiment many times. You find it by multiplying each value by its probability and adding up: a “weighted average”. It needn’t be a value X can actually take (the expected number of tails in 5 flips is 2.5). Its headline use: deciding whether a game is fair — a game is fair when the expected gain is exactly 0.
📘 What you need to know
- E(X) = the expected value (mean) of X — the long-run average.
- Formula: E(X) = ∑ x P(X = x) — multiply each value by its probability, then add. (In the formula booklet.)
- E(X) need not be an obtainable value of X (e.g. 2.5 tails in 5 flips).
- Symmetric distribution: the mean equals the centre value (mean = median) — a quick shortcut.
- Fair game: let X = gain/loss. The game is fair if E(X) = 0.
- Expected gain = (expected prize value) − (cost to play).
- E(X) > 0 → expect a gain; E(X) < 0 → expect a loss.
Calculating E(X)
E(X) is a weighted average: each value pulls on the mean in proportion to its probability.
Expected value of a discrete random variable
E(X) = ∑ x P(X = x)
in the formula booklet ✓
🧭 Recipe — finding E(X)
- Multiply each value x by its probability P(X = x).
- Add all those products together.
- The sum is E(X) — leave it as is, even if it’s not a value X can take.
🤔 Why can E(X) be a value X never takes?
E(X) is an average, not an outcome. A family can’t have 2.4 children, but the average might be 2.4. The expected value tells you what happens “on balance” over many repeats — so a fraction or a value between the possible outcomes is perfectly normal, and you should not round it to a “real” value.
Symmetry shortcut: if the values and their probabilities are symmetric, E(X) is just the middle value — no calculation needed. E.g. X = 1, 5, 9 with probabilities 0.3, 0.4, 0.3 → E(X) = 5.
Fair games
Games are the classic E(X) application. Let X be your overall gain (prize minus cost). The sign of E(X) tells you what to expect.
E(X) = 0
fair
Expected gain is zero — neither player has an advantage in the long run.
E(X) ≠ 0
not fair
Positive → expect a gain; negative → expect a loss. The closer to 0, the fairer.
Expected gain & fairness
expected gain = E(prize) − cost fair ⇔ E(X) = 0
positive = gain, negative = loss
🧠 Memory aid — “prize minus cost”
To judge a game: first find the expected prize with E(X) = ∑xP. Then subtract the cost to play. If what’s left is 0 the game is fair; positive means you profit on average; negative means you lose. Always remember to subtract the cost — forgetting it is the classic slip.
Worked examples
WE 1Calculate an expected value
Daphne wins a prize of $1, $5, $10 or $100. W is the amount won, with probabilities 0.35, 0.5, 0.05, 0.1. Find E(W), the expected prize.
| w | 1 | 5 | 10 | 100 |
| P(W = w) | 0.35 | 0.5 | 0.05 | 0.1 |
E(W) = ∑ w P(W = w)
= 1×0.35 + 5×0.5 + 10×0.05 + 100×0.1
= 0.35 + 2.5 + 0.5 + 10
E(W) = $13.35
multiply each prize by its probability, then add.
WE 2Determine if the game is fair
Daphne pays $15 to play the game in WE 1. Determine whether the game is fair.
expected gain = E(prize) − cost
= 13.35 − 15
= −1.65
is it 0?
expected loss of $1.65 → not fair
E(X) ≠ 0, and it’s negative, so she loses on average.
X takes values 2, 6, 10 with probabilities 0.25, 0.5, 0.25. Find E(X).
spot the symmetry
values 2, 6, 10 are symmetric about 6; probabilities 0.25, 0.5, 0.25 are symmetric too.
E(X) = 6 (the centre)
check: 2×0.25 + 6×0.5 + 10×0.25 = 0.5 + 3 + 2.5 = 6 ✓
A game’s expected prize is $4. What should it cost to play to make the game fair?
fair means E(gain) = 0
E(prize) − cost = 0
4 − cost = 0
cost = $4
a fair game charges exactly the expected prize.
WE 5Expected value with a loss outcome
A spinner gives X = +3, 0, −2 with probabilities 0.2, 0.5, 0.3 (gain in $). Find E(X) and say if it favours the player.
E(X) = ∑ x P(X = x)
= 3×0.2 + 0×0.5 + (−2)×0.3
= 0.6 + 0 − 0.6
E(X) = 0 → fair
expected gain is exactly 0, so neither side has an edge.
💡 Top tips
- E(X) = ∑xP(X = x) — value × probability, then add.
- Don’t round E(X) to an obtainable value — it’s a mean, not an outcome.
- Symmetric distribution → mean = centre value, no working needed.
- Fair game ⇔ E(gain) = 0; remember to subtract the cost to play.
- Sign tells the story: positive E = gain, negative E = loss.
- Include negative outcomes (losses) with their sign in the sum.
⚠ Common mistakes
- Forgetting to subtract the cost when judging fairness — E(prize) alone isn’t the gain.
- Rounding E(X) to a possible value of X. Leave it as the exact average.
- Dropping the sign on negative (loss) outcomes in the sum.
- Adding the probabilities instead of x × P — E(X) is a weighted sum.
- Calling a game fair when E(X) ≠ 0. Fair means exactly 0.
- Using probabilities that don’t sum to 1 — check the distribution first.
That completes the Probability Distributions unit! You can now describe a discrete random variable, find probabilities (including inequalities), and compute the expected value to judge averages and fairness. These ideas extend into the named distributions — the binomial, Poisson and normal — where the same E(X) thinking gives you a mean from a formula rather than a table.
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