Expected Value Formula: Derivation and 5 Worked Examples

Expected value is the long-run average of a random outcome. If you repeated the same uncertain decision an infinite number of times, the average of your results would converge to E[X]. That convergence is guaranteed by the law of large numbers - provided the variance of X is finite.

The formula encodes a simple idea: outcomes that are more likely should count for more, and outcomes that are less likely should count for less. The probability is the weight; the outcome value is what gets weighted. Sum them all up and you have the average you would expect over many repetitions.

Three things to notice. First, the sum runs over every possible outcome - miss one and the formula gives the wrong answer. Second, the probabilities must add to 1, otherwise the calculation is meaningless. Third, expected value can equal a value the variable never actually takes - the expected value of a fair six-sided die is 3.5, which is not a face on the die.

Imagine a game with three possible outcomes: you win £10, you win £0, or you lose £5. Suppose the probabilities are 30%, 50%, and 20% respectively (they sum to 100%, as required). What is the average outcome?

Play the game 100 times. Statistically, you would expect roughly 30 wins of £10 (= £300), 50 nothings (= £0), and 20 losses of £5 (= −£100). Total: £200. Average per game: £2.

The shortcut to that calculation is exactly the expected value formula. The 30 wins of £10 sum to £300, which is the same as 100 × 0.30 × £10. The 20 losses of £5 sum to −£100, which is 100 × 0.20 × (−£5). Add them together, divide by 100, and you get:

E[X] = 0.30 × £10 + 0.50 × £0 + 0.20 × (−£5) = £3 + £0 − £1 = £2

The formula falls out of the long-run average by definition. There is nothing magical about it - it is what "average" means when outcomes happen with different probabilities.

A friend offers you a bet: flip a fair coin. Heads, you win £15. Tails, you lose £10. Should you take it?

Compute the expected value. There are two outcomes, each with probability 0.5.

E[X] = 0.5 × £15 + 0.5 × (−£10) = £7.50 − £5.00 = £2.50

Every time you play, you expect to make £2.50 on average. Over a single flip, you will either gain £15 or lose £10 - those are the only two possibilities. But over a thousand flips, your total winnings should be roughly £2,500. Take the bet.

This is the cleanest illustration of why expected value matters. The bet pays out an asymmetric amount because the probabilities of both outcomes are equal, so the higher payoff dominates. The same expected value calculation generalises to any odds and any payout structure.

The UK National Lottery's Lotto draw requires you to match six numbers from 1 to 59. The probability of matching all six is 1 / C(59, 6) = 1 / 45,057,474. A ticket costs £2 and the jackpot averages around £4 million.

For simplicity, consider only the jackpot outcome and the no-win outcome. The expected value of one ticket is:

E[X] = (1 / 45,057,474) × £4,000,000 + (45,057,473 / 45,057,474) × £0 − £2

E[X] ≈ £0.089 − £2 = −£1.91

Every ticket has an expected value of around negative £1.91. You are paying £2 for an asset worth about 9 pence. Over many years, lottery players lose 96% of what they spend. The other prize tiers (matching five or four numbers) lift the figure slightly but not enough to make the bet positive - under almost any jackpot, the net expected value stays well below the ticket price.

This is one of the cleanest demonstrations that humans are not natural expected-value calculators. We respond to the size of the prize rather than its probability - see the availability heuristic for the psychology behind why lottery wins feel more vivid than the millions of losing tickets that funded them.

You pay £350 a year for buildings insurance on a £300,000 home. The probability of a total loss (fire, flood, structural collapse) in any given year is roughly 0.001 - one in a thousand homes. Smaller claims (storm damage, burst pipes) have higher probability, but assume the policy primarily exists to cover catastrophic loss.

The expected payout to you in any year is:

E[payout] = 0.001 × £300,000 + 0.999 × £0 = £300

Your expected value from buying the insurance is £300 − £350 = −£50. Every year you expect to lose £50 on average. Yet buying the policy is the rational choice - because losing your £300,000 home would be financially ruinous in a way that paying £350 a year is not.

This is the most important practical insight in expected-value thinking. EV is not the whole story when outcomes are non-linear in value - when a £300,000 loss hurts you more than 1,000 times as much as a £300 loss. The framework that handles this properly is expected utility, which weights outcomes by their personal impact rather than their monetary face value.

You are playing No-Limit Hold'em. The pot is £80 and your opponent has bet £40, making the total pot £120. You must call £40 to continue. You estimate (using conditional probability) that your hand has a 35% chance of winning at showdown.

There are two outcomes from calling: you win the £120 currently in the pot, or you lose your £40 call. Apply the formula:

E[call] = 0.35 × £120 + 0.65 × (−£40) = £42 − £26 = £16

The call has positive expected value of £16. Make the call. Every time you face this exact situation, you will sometimes win and sometimes lose, but the average across many such hands will be a £16 gain per call.

This is the calculation behind every poker decision worth taking seriously. The crucial input is your equity estimate - the 35% above. Get that wrong and the formula gives the wrong answer. The skill in poker is partly about working out your equity quickly and honestly; once you have it, the expected-value arithmetic is mechanical. See expected value in poker for more applications.

You are considering a £10,000 investment in a single stock. Three scenarios:

Apply the expected value formula:

E[final value] = 0.30 × £14,000 + 0.50 × £10,800 + 0.20 × £7,000

E[final value] = £4,200 + £5,400 + £1,400 = £11,000

The expected final value is £11,000, so the expected return is (£11,000 − £10,000) / £10,000 = 10%. On expected-value grounds, the investment looks attractive - provided the input probabilities and outcome values are honest estimates.

But expected value alone does not capture risk. The same expected return is available from a fixed-income product with no bear-case loss, and most rational investors would prefer that. The full framework needs to account for the distribution around the expected value, which is what variance and Sharpe ratio capture. For position sizing on an investment like this, see the Kelly Criterion calculator.

Expected value is the right number to maximise across a long sequence of independent, repeatable bets where the outcomes are linear in their personal impact. Three situations break those assumptions and require a different framework.

One-shot decisions with ruinous downside. If a single outcome would wipe you out (financially, professionally, medically), you should not maximise EV. The insurance example above is the classic case - the expected value of buying the policy is negative, but the policy is rational because losing the house is not 1,000 times as bad as losing the premium, it is infinitely worse. Expected utility handles this case.

Non-ergodic systems. Some processes have a time-average that diverges from the ensemble-average. The classic example is a 50% gain followed by a 50% loss: on average across many traders the wealth stays flat, but a single trader playing repeatedly trends toward zero. See ergodicity explained for why this matters for compounding investments.

Adversarial settings with shared information. If you are taking a market price as input to your EV calculation, the price already encodes other people's EV estimates. Acting on EV alone in markets means systematically betting against people with better information than you - see risk versus uncertainty for the distinction Knight drew between the two in 1921.

• Expected value explained - the conceptual pillar
• Expected value calculator - apply the formula in a worksheet
• Expected value vs expected utility - when EV is the wrong framework
• Kelly Criterion calculator - sizing positive-EV bets
• Law of large numbers - why long-run EV is reliable