Expected Value Explained
Expected value is the single most important concept in gambling mathematics. It tells you, on average, how much you will win or lose per bet — and why the house always wins in the long run.
What Is Expected Value?
Expected value (EV) is the average outcome you would get if you repeated a bet an infinite number of times. It is calculated by multiplying each possible outcome by its probability and adding the results together.
A positive EV means you profit on average. A negative EV means you lose on average. Every casino game is designed to have a negative EV for the player.
Step-by-Step Examples
Example 1: European Roulette (Single Number Bet)
You bet $10 on a single number. There are 37 pockets (0–36).
- • Probability of winning: 1/37 ≈ 2.70%
- • Payout on win: $350 profit (35:1)
- • Probability of losing: 36/37 ≈ 97.30%
- • Loss: $10
Every $10 bet costs you $0.27 on average. The house edge is 2.70%.
Example 2: American Roulette (Red/Black Bet)
You bet $10 on Red. There are 38 pockets (0, 00, 1–36). 18 are red.
- • Probability of winning: 18/38 ≈ 47.37%
- • Payout on win: $10 profit (1:1)
- • Probability of losing: 20/38 ≈ 52.63%
- • Loss: $10
American roulette has a 5.26% house edge — nearly double European roulette — because of the extra 00 pocket.
Example 3: A Fair Coin Flip (Zero House Edge)
You flip a fair coin for $10. Heads you win $10, tails you lose $10.
This is a zero-EV (fair) game. Casinos never offer this — there is always an edge built in through the payout structure.
Why EV Matters More Than Individual Results
You can win on any single bet, even with terrible odds. EV does not predict individual outcomes — it predicts the average over thousands of repetitions. This is the core insight:
- •Short term: Variance rules. Anything can happen on any given session.
- •Long term: EV rules. Results converge to the mathematical expectation.
- •No system changes EV: Changing bet sizes or sequences does not alter the underlying house edge.
A gambler who wins big on their first visit has not beaten the house — they have experienced favorable short-term variance on a bet with negative EV. Play long enough and the math takes over.
How to Use the Expected Value Calculator
- 1.Enter your win probability. For a coin flip this is 50%. For a roulette single number it is 1/37 ≈ 2.70%. Look up the odds for your specific game.
- 2.Enter your net win amount. If you bet $10 and win $350 back, your net win (profit) is $350. Do not include the return of your stake.
- 3.Enter your net loss amount. This is your stake — the amount you lose when the bet fails, e.g., $10.
- 4.Read the result. A negative EV (shown in red) means the bet costs you money on average. The house edge percentage tells you how much of every dollar wagered the casino keeps.
Common House Edges at a Glance
| Game | Bet Type | House Edge |
|---|---|---|
| European Roulette | Any bet | 2.70% |
| American Roulette | Any bet | 5.26% |
| Blackjack | Basic strategy | 0.5% |
| Baccarat | Banker bet | 1.06% |
| Slot Machines | Any | 2–15% |
| Craps | Pass line | 1.41% |
Try the Expected Value Calculator
Enter any bet to see exactly what it costs you on average.