The average you expect
Expected value is the average result you’d get if you repeated a random process many times.
E(X) = Σ x × P(x)
It’s a weighted average: each value is weighted by its probability.
Example: a die
E(X) = (1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5
You can’t roll 3.5, but over many rolls, the average approaches 3.5.
Games and payoffs
Example: Pay $2 to play. Roll a die: win $10 for a 6; otherwise nothing.
| Outcome | Net payoff | Probability |
|---|---|---|
| roll 6 | +$8 | 1/6 |
| other | −$2 | 5/6 |
E = 8(1/6) + (−2)(5/6) = 8/6 − 10/6 = −$0.33
On average you lose about 33 cents per game in the long run.
Fair games
A fair game has E = 0. To make the game above fair, find the prize W:
(W − 2)(1/6) + (−2)(5/6) = 0 → W − 2 = 10 → W = $12
Lotteries and casinos
Games of chance are designed with a negative expected value for players. The house edge is the casino’s expected profit per dollar bet.
Example: In American roulette, betting $1 on red: 18 red, 18 black, 2 green.
E = (+1)(18/38) + (−1)(20/38) = −2/38 ≈ −$0.053
House edge ≈ 5.3%.
Insurance
Example: A phone insurance plan costs $90/year. P(the phone is lost or broken) = 0.05; replacement costs $1,000.
- Expected cost without insurance: 0.05 × 1,000 = $50
- Insurance costs $90, more than $50 on average
People still buy insurance to avoid a large, unpredictable loss. Expected value isn’t the only factor in decisions.
Business decisions
Example: A product launch has a 40% chance of earning $500,000 and a 60% chance of losing $200,000.
E = 0.4(500,000) + 0.6(−200,000) = 200,000 − 120,000 = $80,000
Positive expected value suggests launching, but risk tolerance matters.
Binomial shortcut
For a binomial variable, E(X) = np. 50 free throws at 80% → expect 40 made.
Worked examples
Example 1: X = 0, 5, 10 with P = 0.5, 0.3, 0.2. E(X)? 0 + 1.5 + 2 = 3.5.
Example 2: A raffle sells 1,000 tickets at $5; one prize of $2,000. Expected value of a ticket? (1,995)(0.001) + (−5)(0.999) = −$3.
Example 3: Is a game with E = $0.25 favourable to the player? Yes (on average).
Common mistakes
- Forgetting the cost to play when finding net winnings.
- Expecting the expected value in a single game.
- Probabilities that don’t add to 1.
Try it yourself
- A spinner: win $4 with P = 0.25, else win $0. Cost to play $1. Expected net value?
- Values 1, 2, 3 with P = 0.2, 0.5, 0.3. E(X)?
- What prize makes a $1 game with a 1/10 chance of winning fair?
- 30% of shots score. Expected goals in 20 shots?
Answers: 1) $0 (fair) 2) 2.1 3) $10 total payout (net +$9) 4) 6
Quick recap
- Expected value = Σ x P(x), a weighted average.
- A fair game has E = 0; casinos have a house edge.
- Expected value describes the long run, not one trial.