Mental models

Expected Value

The probability-weighted average that tells you which bets to take — and the four situations where following it misleads you.

Expected value is the probability-weighted average payoff of a decision, calculated by multiplying each outcome's payoff by its probability and summing those products over a complete outcome set whose probabilities total 1 (EV = Σ probability × payoff).

TL;DR

  • Expected value (EV) is the probability-weighted average of a decision's possible outcomes: EV = Σ(probability × payoff). Multiply each outcome by its chance of happening, then add the products. The probabilities must cover every outcome and sum to 1, and every payoff must be measured in a single unit such as dollars.
  • A single-number bet in American roulette has negative expected value: EV = (1/38 × +$35) + (37/38 × −$1) = $0.9211 − $0.9737 = −$0.0526 per dollar staked. The house edge of 5.26 percent is exactly this arithmetic, and no betting system alters it.
  • Positive expected value does not make a bet safe. EV describes the average of many independent repetitions, so it says almost nothing about a single decision, and nothing at all about a sequence in which one bad draw wipes out the stake. Survival comes first: you must stay in the game to collect the average.
  • Expected utility replaces dollars with the value those dollars have for the decider. Because an extra dollar usually adds less than the one before it, rational agents maximize probability-weighted utility rather than probability-weighted cash. That is why people buy insurance with negative monetary EV, and why the infinite-EV St. Petersburg gamble is worth only a few dollars.
  • Bet sizing turns positive EV into growth. When a loss costs the whole stake, the Kelly criterion stakes fraction f = (bp − q)/b of the bankroll, where b is the net odds, p the win probability, and q = 1 − p. An even-money bet won 60 percent of the time takes 20 percent, not everything.

When to use it

  • Use expected value when the same kind of decision repeats many times and each individual loss is survivable: pricing insurance policies, buying ad inventory, stocking products. Across many trials the realized average converges on the EV by the law of large numbers, so the highest-EV option wins over the long run.
  • Use it to compare options with different odds and different stakes, because EV reduces them to one comparable number. A 5 percent shot at $200,000 has EV $10,000 and beats a certain $8,000 for a firm that can absorb the loss, even though it pays nothing 95 percent of the time.
  • Use it to expose bad bets that feel good. Lotteries, most parlays, and extended warranties carry negative EV that a vivid jackpot or a vivid disaster hides. Writing out Σ(probability × payoff) turns the story into a number, and when the loss is one you could absorb yourself, the negative number settles it.
  • Use it as the engine of a decision tree, folding values back to the root. Give each terminal node its payoff, take the probability-weighted average at every chance node, take the maximum at every decision node, then carry the value upward. Root values rank whole strategies rather than isolated choices.

When it fails

  • Expected value fails when a loss is unrecoverable. A bet with a positive average is still wrong if one outcome ends the game — bankruptcy, death, a destroyed reputation — because you never get to replay the average. Risk of ruin, not EV, governs any decision whose downside removes you from all future decisions.
  • Expected value fails when you read it as a forecast of one decision, because the average is often an outcome that cannot occur. A fair coin flip paying $0 or $1,000 has EV $500, yet a single play returns $0 or $1,000, never $500. EV ranks repeatable options; it predicts no individual result.
  • Expected value fails when payoffs are not linear in money. Losing your last $10,000 hurts far more than gaining $10,000 helps, so expected utility rather than expected dollars describes rational choice. This is why buying insurance, which has negative monetary EV by construction, is still a sensible purchase.
  • Expected value fails when the probabilities are guesses. Multiplying an invented 3 percent by an invented $50 million produces a confident-looking $1.5 million that is pure fiction. Under genuine uncertainty — no reliable base rate, fat-tailed outcomes — EV inherits every error in its inputs and disguises them as precision.

Worked example

A company weighs a product launch with three possible outcomes. A hit, probability 0.10, earns $900,000. A modest success, probability 0.30, earns $100,000. A flop, probability 0.60, loses $150,000. The probabilities sum to 1.00, so the outcome set is complete. Multiply each payoff by its probability: 0.10 × $900,000 = $90,000; 0.30 × $100,000 = $30,000; 0.60 × (−$150,000) = −$90,000. Adding the three products gives EV = $90,000 + $30,000 − $90,000 = +$30,000. The launch is worth doing on EV grounds, but notice that it loses money 60 percent of the time — the entire positive average comes from the rare hit.

Roulette runs the same arithmetic in the other direction. An American wheel has 38 pockets, and a single-number bet pays 35 to 1. Staking $1: EV = (1/38 × +$35) + (37/38 × −$1) = $0.9211 − $0.9737 = −$0.0526. Every dollar wagered loses about 5.3 cents on average, the familiar 5.26 percent house edge, so 10,000 one-dollar spins carry an expected loss of about $526. The casino, not the gambler, is the party running the repeated experiment, and that asymmetry is the whole business model.

Positive expected value can still grind a stake to nothing if you bet too much of it. Consider a fair coin flip that doubles your money on heads and takes 60 percent of it on tails: EV per round = 0.5 × (+100%) + 0.5 × (−60%) = +20%, which looks irresistible. But wealth multiplies rather than adds. Betting everything each round, one head and one tail in either order leaves 2.0 × 0.4 = 0.8 of the stake, and after n such pairs 0.8^n, which converges to zero. The Kelly criterion fixes the size, not the edge. For a gamble that gains fraction g or loses fraction l of the amount staked, f = (pg − ql)/(gl), so here f = (0.5 × 1 − 0.5 × 0.6)/(1 × 0.6) = 1/3. Stake a third and the same edge compounds instead of decaying.

The St. Petersburg paradox breaks expected value outright. A fair coin is flipped until it lands heads. The pot starts at $2 and doubles with every tail, so the payoff is $2^k when the first heads appears on flip k, an event with probability 1/2^k. Every term contributes 1/2^k × $2^k = $1, and there are infinitely many terms, so the expected value is infinite. Yet almost nobody will pay more than a modest sum to play. Nicolas Bernoulli posed the puzzle in 1713; Daniel Bernoulli's 1738 answer was expected utility: a player who values a prize of x dollars at ln x has expected utility 2 ln 2 = ln 4, a certainty equivalent of exactly $4.

Related concepts

  • Sunk Cost Fallacy — Expected value counts only future probability-weighted payoffs; the sunk cost fallacy is the error of letting already-spent money enter that calculation.
  • Nash Equilibrium — Expected value is the machinery underneath mixed-strategy equilibria, where each player's mix leaves the opponent indifferent because the mixed-in options carry equal expected payoffs.
  • Dominant Strategy — A dominant strategy needs no probabilities at all, since it pays best in every state — expected value is what you compute when no such strategy exists.
  • Prisoner's Dilemma — Defection dominates in a one-shot game, so it wins on expected value against any belief about the other player; repetition changes the payoff stream, not the arithmetic.

FAQ

What is expected value in simple terms?

Expected value is what a decision pays on average if you could repeat it many times. Multiply each possible result by its probability, then add everything up. A $1 bet that pays $10 one time in five has EV = (0.2 × $10) + (0.8 × $0) = $2 against a $1 stake, a net +$1.

How do you calculate expected value?

Calculate expected value with EV = Σ(probability × payoff). List every possible outcome, assign each a probability so the list sums to 1, multiply each probability by that outcome's payoff, then add the products. Worked example: 0.25 × $400 + 0.75 × (−$100) = $100 − $75 = +$25.

When does expected value fail?

Expected value fails when one bad outcome is unrecoverable, when payoffs are not linear in money, or when the probabilities are invented. It also predicts no single trial: EV is a long-run average, so a one-shot decision returns an actual outcome, never the average, and inherits every error in its inputs.

What is the difference between expected value and expected utility?

Expected value averages payoffs in objective units such as dollars; expected utility averages the value those payoffs carry for the decider. Because utility rises more slowly than wealth, the two rank options differently. Insurance lowers expected wealth but raises expected utility, and the infinite-EV St. Petersburg gamble has a small, finite utility value.

Can a positive expected value bet still be a bad idea?

Yes. A positive-EV bet is a bad idea when the stake is too large to survive losing. Betting an entire bankroll on a 60 percent even-money edge has EV of +20 percent per round, yet ruin becomes near-certain over time. The Kelly criterion sizes that same edge at 20 percent of bankroll.

Why do casinos and insurers rely on expected value?

Casinos and insurers rely on expected value because they, not their customers, run the repeated experiment. A single-number roulette bet loses the player 5.26 cents per dollar; across millions of spins the house's realized take converges on that figure, while an individual player's handful of spins is dominated by variance.