What a raffle ticket is really worth
A charity sells raffle tickets for $2. The prizes and odds are printed on the back, but the real question is simple: on average, what does one ticket return? The expected value answers exactly that. It is the long-run average outcome if you repeated the gamble many times.
The expected value formula
Formula: E(X) = sum of [x x P(x)] for each possible outcome.
Multiply each outcome x by its probability P(x) and add the results. The probabilities must add up to 1. The result is a weighted average where likely outcomes count more than rare ones.
Using the calculator
- Enter each possible outcome and its probability as a decimal.
- Check that the probabilities sum to 1.
- Read the expected value in the same units as the outcomes.
A worked example, step by step
A game pays $0, $1, $2, or $5 with probabilities 0.4, 0.3, 0.2, and 0.1.
- Step 1: check the probabilities: 0.4 + 0.3 + 0.2 + 0.1 = 1.0.
- Step 2: weight each outcome: 0 x 0.4 = 0, 1 x 0.3 = 0.3, 2 x 0.2 = 0.4, 5 x 0.1 = 0.5.
- Step 3: add them: 0 + 0.3 + 0.4 + 0.5 = 1.2.
The expected value is $1.20 per play. A $2 ticket is a losing deal on average.
Where expected value matters
Insurers price policies with it, investors compare strategies with it, and game designers balance payouts with it. To see how much results vary around the average, use the variance calculator and the standard deviation calculator, or start from raw chances with the probability calculator.
Mistakes that give the wrong answer
- Probabilities that do not sum to 1. Using percentages like 40 instead of 0.4, or an incomplete list of outcomes, breaks the weighting.
- Treating the expected value as a promise. You will never win exactly $1.20 in one play; it is the average over many plays.
- Ignoring the spread. Two games can share an expected value while one is far riskier, so always check the variance too.
The house edge is negative expected value
Casino games are engineered so the player expected value sits slightly below zero. A roulette bet on red pays even money with a win probability of 18/37 on a European wheel, giving an expected return of (1 x 18/37) + (-1 x 19/37) = -1/37, about minus 2.7 cents per dollar bet. That small negative number, multiplied by millions of bets, is the entire business model. Whenever someone offers you a game, computing the expected value tells you exactly who the game was built for.
What this assumes
Expected value describes the long run. In a handful of trials, actual results can land far from it, and it says nothing about risk on its own.