Expected Value Explained: Roulette, Insurance, and Everyday Decisions
October 21, 2026
Expected value (EV) is the average result you would get if you repeated a random situation many times. You compute it by multiplying each possible outcome by its probability and adding the results. It is the foundation of decision-making under uncertainty in games, finance, insurance, and business. A positive EV means the bet is favorable on average, and a negative EV means you lose on average.
In American roulette, a wheel has 38 pockets: numbers 1 to 36 (18 red, 18 black) and two green zeros. A $1 bet on red wins $1 with probability 18/38 and loses $1 with probability 20/38. EV = (18/38) × 1 + (20/38) × (−1) = −2/38 = −$0.0526. The player loses about 5.26 cents per dollar bet, on average. That is the house edge.
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Open the Average Calculator →Computing Expected Value
Steps: list each outcome and its payoff, multiply by the probability, and sum. A single-number bet at 35 to 1: the chance of winning is 1/38 for +$35 and 37/38 for −$1, giving EV = 35/38 − 37/38 = −0.0526, the same house edge. All standard roulette bets share this edge in the American version. European roulette has one zero (37 pockets), which reduces the edge to 2.7%.
The EV is a long-run average. In a single spin you win or lose $1, never −$0.0526. Over 1,000 spins of $1 you expect to lose about $52.60, though the actual result will vary around that.
| Bet (American roulette) | Win probability | Payout | Expected value per $1 |
|---|---|---|---|
| Red or black | 18/38 = 47.37% | 1 to 1 | -$0.0526 |
| Single number | 1/38 = 2.63% | 35 to 1 | -$0.0526 |
| Dozen | 12/38 = 31.58% | 2 to 1 | -$0.0526 |
Insurance and Negative EV
Insurance has a negative expected value for the buyer, because insurers must cover costs and profit. If a policy costs $500 a year and the expected claim payout is $350, EV = −$150. Yet buying it can be rational, because the goal is not to maximize expected value but to avoid ruin. A fire that destroys your uninsured home is a loss you cannot absorb, so paying a small certain premium to remove a large uncertain loss improves your situation.
This reflects diminishing marginal value of money: losing $300,000 hurts more than 600 times as much as paying $500.
Business and Investment Decisions
A company considers a project that costs $100,000 with a 30% chance of generating $400,000, and a 70% chance of generating nothing. EV = 0.3 × 400,000 − 100,000 = $20,000. A positive EV suggests it is attractive, but the company should consider whether it can survive a failure and whether the probabilities are realistic. Repeated bets of this type across many projects make EV more meaningful, since the law of large numbers applies.
Venture investors use this logic: a few big winners pay for many losers.
Limits of Expected Value
Probabilities are often unknown or poorly estimated. EV ignores risk preferences, so two options with the same EV can feel very different. The St. Petersburg paradox shows a game with infinite expected value that no reasonable person would pay much to play. Rare catastrophic outcomes require special care. In one-off decisions, you do not get the long-run average, and the variance matters.
Using EV Day to Day
Compare extended warranties, lottery tickets, and promotions by their expected value. Most extended warranties have negative EV for the buyer. A coupon that requires a $50 spend to save $10 may have positive EV if you were going to spend that amount anyway, and negative if you would not.
Frequently Asked Questions
What is expected value?
The probability-weighted average of all possible outcomes.
What is the house edge in American roulette?
5.26% for almost all bets.
Why do people buy insurance if it has negative EV?
To avoid large losses they could not absorb, accepting a small certain cost.
Is a positive EV bet always a good bet?
Not if the downside could ruin you or your estimates are unreliable.
How do I calculate EV?
Multiply each outcome's value by its probability and add them up.