Statistics Reference
Expected Value Explained
Expected value is the single number that summarizes a gamble: the average result per play if you could repeat it forever. It prices raffle tickets, sets insurance premiums, and quietly runs every casino. This guide computes it three ways on real examples, then marks the boundary where “positive EV” stops being the whole answer.
The Formula
E(X) = Σ xᵢ × P(xᵢ)
Fair die: E(X) = (1+2+3+4+5+6)/6 = 3.5
Each outcome contributes in proportion to its probability — a weighted average with probabilities as the weights (the same arithmetic as the weighted mean, with weights that must sum to 1). The die's 3.5 illustrates the key subtlety: E(X) is not a possible outcome, it is where the long-run per-roll average lands. Roll the die 10,000 times and the mean of your results will sit within a whisker of 3.5 — that convergence of sample mean to expected value is the law of large numbers.
Worked Example 1: Pricing a Raffle Ticket
A charity sells 1,000 tickets at $5; one wins a $2,000 prize. Net payoffs per ticket:
| Outcome | Net payoff | Probability | Contribution |
|---|---|---|---|
| Win | +$1,995 | 0.001 | +$1.995 |
| Lose | −$5 | 0.999 | −$4.995 |
| E(X) | −$3.00 | ||
Each ticket loses $3 on average — precisely the gap between the $5 price and the $2 of prize money spread across 1,000 tickets. The shortcut view: total prize / total tickets − price = 2000/1000 − 5 = −3. Entering the two outcome/probability pairs into the expected value calculator reproduces the result along with the variance and standard deviation of the gamble.
Worked Example 2: Dice, Coins, and Repeat Play
A game pays the face value of one fair die roll, and costs $4 to play. E(payout) = 3.5, so E(net) = −$0.50 per play: a slow leak. Over 100 plays the expected total is −$50 — and multi-event tools make the same logic concrete:
- The expected sum of two dice is 3.5 + 3.5 = 7 (expectations add, even for dependent events) — compare the exact sum distribution in the dice probability calculator.
- The expected number of heads in 10 fair flips is np = 5 — the coin flip calculator shows the full distribution around that center.
- An event with per-trial probability p occurs np times on average across n tries; the at-least-one calculator separates that expected count from the probability of at least one occurrence — two numbers beginners often conflate.
Worked Example 3: Why Insurance Works for Both Sides
A policy costs $500/year and pays $10,000 for a loss that strikes 2% of customers annually.
- Insurer's EV per policy: +500 − 0.02 × 10,000 = +$300. Across thousands of policies, the law of large numbers turns that average into near-certain revenue.
- Buyer's EV: −$300. Rational anyway: the buyer trades a small certain cost for protection against a loss they might not absorb. What insurance maximizes is not expected dollars but expected utility — the value of money when you need it most.
The same asymmetry explains why casinos welcome high volume while individual gamblers should fear it: repetition locks the average in for whoever holds the positive EV.
Where EV Alone Falls Short
- Variance and ruin. Two bets with identical EV can carry wildly different spreads; if a bad streak can bust you, the long run never arrives. The expected value calculator's variance output is the companion number to check.
- One-shot decisions. The long-run average is a weak guide to a decision you make once — the distribution of outcomes, not its center, is what you experience.
- Nonlinear stakes. When the 100th dollar and the 100,000th dollar differ in value to you (they do), expected utility, not expected money, is the right maximand — the principle behind both buying insurance and declining positive-EV bets that risk too much.
Try the Expected Value Calculator
Enter outcomes and probabilities to get E(X), variance, and standard deviation — every worked example in this guide, automated.
Frequently Asked Questions
What is expected value in simple terms?
It is the probability-weighted average of all possible outcomes: E(X) = sum of (each outcome x its probability). It answers 'what would I average per play if I repeated this many times?' A fair die has E(X) = 3.5 - not a value you can ever roll, but exactly what the per-roll average approaches over many rolls.
How do I calculate the expected value of a game or bet?
List every outcome with its net payoff and probability, multiply each pair, and add. A $5 raffle ticket in a 1,000-ticket draw for a $2,000 prize: E = (1/1000)(+1995) + (999/1000)(-5) = 1.995 - 4.995 = -$3. On average each ticket loses three dollars, which is the organizer's margin.
Why is the expected value often a number that can never actually happen?
Because it is an average over outcomes, not a prediction of one outcome. A die never shows 3.5 and a household never has 2.3 children. Expected value describes the long-run per-trial average (and the center of the distribution), not what any single trial will produce.
If a bet has positive expected value, should I always take it?
Not automatically. EV describes the long-run average; it says nothing about the spread or about your capacity to survive the short run. A positive-EV bet that risks your entire bankroll can still be ruinous (variance matters, and repeated all-in bets eventually bust), and a negative-EV purchase like insurance can be rational because it removes a catastrophic outcome. EV is one input to a decision, not the decision.
How is expected value related to the mean?
They are the same idea in two dresses: the mean averages data you have observed; the expected value averages a probability model's outcomes before observing anything. The law of large numbers is the bridge - as trials accumulate, the sample mean of results converges to the model's expected value.
Why does insurance have negative expected value for the buyer, and people still buy it?
The premium must exceed the expected claim payout or the insurer could not operate - in the worked example the buyer's EV is -$300 per year. Buying is still rational because the buyer is not maximizing EV in dollars: they are removing a small probability of a catastrophic loss they could not absorb. Utility - the value of money to you - is what insurance optimizes.