The average outcome — and why that is rarely the outcome.
The average, and the spread it hides
Name, probability, payoff. Probabilities must sum to 1. A single number on a European wheel. The house edge is exactly −1/37.
Expected value over 2 outcomes
-0.027027
Standard deviation 5.837838 — far larger than the expected value, so a single play tells you almost nothing about it.
Expected value
-0.027027
Standard deviation
5.837838
Chance of any gain
2.7027%
Chance of a loss
97.2973%
The single most likely outcome is “Lose” at 97.2973%, paying -1.00000. Here that sits close to the expected value, which is the comfortable case rather than the usual one.
The spread is 216.0000 times the expected value. At that ratio the expected value is nearly useless as a guide to one play. You would need roughly 46,656 repetitions before the average outcome reliably resembles it — because the standard error of the mean falls only as the square root of the number of plays.
A fair price of 10.00000, over at most 20 tosses. The St Petersburg game pays 2ⁿ⁻¹ when a fair coin first lands heads on toss n. Every term contributes exactly one half, so the expected value diverges: it is infinite, and almost nobody will pay twenty for it. The resolution is not psychology. A real counterparty cannot pay an unbounded prize, and once the bank is finite the sum stops. A bank of a million caps the fair price at 10; a billion at 15. A thousandfold richer opponent is worth five more — the value grows with the logarithm of the bankroll, which is why the infinity was always an artefact of an assumption nobody could honour.
What this tool shows
An expected value quoted alone is half a description. A roulette number bet has an EV of −2.70% and a standard deviation of 5.84, and its single most likely result is losing everything — 36 times in 37. This reports the spread, the chance of any gain, and the modal outcome alongside.
Expected value from any set of outcomes
Standard deviation, which the average alone hides
The chance of any gain, and of a loss
The single most likely outcome
How many repetitions before the average is meaningful
St Petersburg, and what a finite bankroll does to it
Spread reported Modal outcome shown Probabilities checked St Petersburg included
A roulette bet needs about 46,657 spins before the average resembles its EV.
Updated 8 September 2026 · Works in any browser, no installation
Expected value is each outcome multiplied by its probability, summed. It is the average you would converge on over many repetitions — which makes it the right basis for a decision you will make many times, and a poor one for a decision you will make once.
At a glance
Formula shown
EV = Σ pᵢ · xᵢ, with the probabilities summing to 1. Variance = Σ pᵢ(xᵢ − EV)², and its square root is the spread the average conceals. The number of repetitions before the observed mean is reliably close to the EV grows as the square of the ratio between them, because the standard error falls only as √n.
Scenario support
Comparing bets, insurance, or business options; pricing a risk; deciding whether a positive-EV opportunity is actually takeable; understanding a house edge; any choice where outcomes have known probabilities.
Educational estimate
Planning support from the values you enter — not professional advice.
The expected value is often an outcome that cannot happen
This is the most useful thing to know about expected value, and it follows directly from the definition rather than being a caveat bolted on.
A single-number roulette bet has an expected value of −2.70% per unit staked. No spin ever pays −0.027. You either lose 1 or win 35. The expected value is a property of the distribution, not a member of it.
The modal outcome — the single likeliest result — is losing everything, 36 times in 37. The tool reports it, because “the average is −2.70%” and “you almost certainly lose” are both true and only one of them is usually said.
The gap is measured by the standard deviation, and here it is 5.84. That is 216 times the size of the expected value. At that ratio a single play tells you essentially nothing about the expectation, and the tool says how many plays it would take: about 46,657, because the standard error of the mean falls only as the square root of the number of repetitions.
Which is exactly why casinos are profitable and gamblers are not. The house plays millions of spins and lands reliably on its 2.70%. An individual plays a few hundred and lands somewhere in a wide distribution around it. Same expected value, completely different exposure — and the difference is the repetition count, not the odds.
So the practical rule is about how many times you will face the decision. Expected value is close to sufficient for a choice repeated thousands of times. For a decision made once, with a spread far larger than the mean, it is one input among several rather than the answer.
Why a negative expected value can be the right choice
“Take the positive-EV option” is a good default and not a rule, and insurance is the clearest counterexample.
Every insurance policy has a negative expected value for the buyer. It has to: the insurer pays claims, wages and profit out of the premiums. The tool’s insurance preset has an EV of −85 and buying it is entirely rational.
The reason is that money is not linear in usefulness. Losing £250 in premiums is an inconvenience; losing £8,000 uninsured might be a catastrophe you cannot absorb. The second is more than 32 times worse than the first, even though it is 32 times larger. Expected value weights outcomes by their monetary size, which assumes the two scale together.
Expected utility is the formal repair — apply a concave function to the payoffs before averaging, so large losses count for more than proportionally. It also explains why the same person buys insurance and a lottery ticket without being inconsistent.
The other reason is ruin. A bet with a positive expected value can still be a bad idea if losing ends the game: an outcome you cannot come back from removes every future repetition, and expected value assumes those repetitions exist. The Kelly criterion is the standard answer — it maximises the long-run growth rate rather than the expectation, and it never stakes everything however favourable the odds.
Load the lottery preset for the mirror image. It has a genuinely positive expected value of +1.10 — which happens in real rollover jackpots — and a 2% chance of any gain at all. Positive EV is not the same as a good bet, in either direction.
The infinite expected value that is worth about ten
The St Petersburg game is the sharpest demonstration that an expected value can be technically correct and practically meaningless.
The game: a fair coin is tossed until it lands heads. If the first head is on toss n, the payoff is 2n−1. What is a fair price to play?
The expected value is infinite. The probability of the first head on toss n is 1/2n, and the payoff is 2n−1, so every term contributes exactly one half. Add infinitely many halves and the sum diverges. Formally, no price is too high.
Almost nobody will pay twenty. That was the paradox as Daniel Bernoulli posed it in 1738, and it was taken for two centuries as evidence that people are irrational about money.
The resolution is arithmetic, not psychology: no real counterparty can pay an unbounded prize. Once the bank is finite the sum stops, and the tool computes where. A bank of one million caps the fair price at 10. A billion caps it at 15. A counterparty a thousand times richer is worth five more.
The value grows with the logarithm of the bankroll, so it never reaches anything like infinity for any bank that could exist. The refusal to pay twenty was correct all along, and the model was wrong — it assumed a counterparty nobody has ever met.
The general lesson is worth more than the puzzle. An expected value computed over a tail nobody can actually pay is not a fair price. The same failure appears in any risk model whose extreme outcomes exceed the capital of the party on the other side.
Four errors the arithmetic will not catch
The formula is a weighted sum, so it computes cleanly on inputs that are wrong. These are the four ways they usually are.
Probabilities that do not sum to 1. Usually a missing outcome. The tool reports the sum and refuses to normalise silently, because rescaling would hide the omission rather than reveal it — and the missing outcome is often the one that matters.
Payoffs stated on the wrong basis. A roulette number pays 35 to 1, meaning you receive 35 plus your stake back. Entering 36 as the payoff while also entering −1 for a loss double-counts the stake and turns the house edge positive. Decide whether payoffs are net or gross and keep every row consistent.
Outcomes that are not mutually exclusive. Expected value sums over a partition. If two listed outcomes can both occur, the weights are not probabilities of distinct events and the sum means nothing — the same requirement the probability calculator enforces for its four-cell breakdown.
Probabilities that were guessed. The output has exactly the authority of its inputs, and a precise-looking expected value built on estimated probabilities inherits their uncertainty without displaying it. Where the probabilities come from data, the beta distribution gives an interval for them; where they are judgements, vary them and see whether the decision changes before trusting the number.
Where it is genuinely decisive
The criticisms above are about relying on it alone. Used with its spread it is one of the most useful quantities in applied probability.
Anything repeated at scale. Pricing, underwriting, insurance, insurance reserving, advertising spend, inventory decisions. When the same decision is made ten thousand times, the average is what you will actually get, and the spread on the total shrinks even as the spread per instance does not.
Comparing options on one axis. Two projects with different payoff structures become comparable once each is reduced to an expectation, which is what makes it the backbone of decision analysis and of expected monetary value in project management.
Detecting a house edge. Any game with a negative expectation cannot be beaten by staking strategy. A martingale — doubling after each loss — does not change the expected value at all; it converts a small frequent loss into a rare enormous one, which feels different and is not better.
Valuing information. The expected value of a decision with perfect information, minus its expected value without, is exactly what that information is worth — and it puts a ceiling on what any amount of research or testing can be worth paying for.
The habit to build is to report both numbers. “Expected value 1.10, standard deviation 10,696” is a complete description; the first number alone is an invitation to the mistake this page is about.
Method. The expected value and variance are computed from the outcomes exactly as entered, and the probability sum is reported rather than normalised — a set summing to 1.4 is a missing or double-counted outcome, and rescaling it would conceal that. The tool also reports the modal outcome and the chance of any gain, because on a negative-expectation bet those two facts carry the part of the story the average removes. The repetition estimate comes from the square of the spread-to-mean ratio, which is where the standard error of the mean reaches the size of the effect. The suite asserts that a fair coin bet has an expected value of exactly zero with a standard deviation of exactly 1, that a single-number roulette bet returns exactly −1/37 per unit with the modal outcome a loss, that probabilities summing to 1.4 are flagged invalid rather than accepted, and that St Petersburg’s fair price is 10 against a bank of a million and 15 against a billion — logarithmic growth, five per thousandfold, not the infinity the unbounded formulation implies. That engine is verified on every change against 99 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
ProbabilityTwo events, repeated trials and Bayes, with the three usual errors handled — the dropped overlap in P(A or B), n×p instead of the complement, and the base rate that makes a 99% test 17% right.
Beta DistributionTakes raw successes and failures and runs the conjugate update, so 5 out of 5 returns 6/7 rather than the 100% a plain proportion claims — with a credible interval that stays honest at zero.
Binomial DistributionExact binomial probabilities at any n — including thousands, where a factorial overflows — with the normal approximation beside them and its error measured, which is 0.6% at the centre and 261% in the tail.
Monty HallGeneralises the puzzle so the famous 100-door explanation stops hiding its own assumption — open one door instead of 98 and switching is worth 1.0102% against 1%, still right and barely.
Odds RatioOdds ratio, relative risk, risk difference and number needed to treat from one 2x2 table — because an odds ratio of 6.00 can describe a relative risk of 1.50.
Weighted AverageEach value carries the weight you give it, with every item's share of the total shown as a percentage so you can see what is actually driving the answer.
An educational tool, not financial or gambling advice. Expected value assumes the decision will be repeated enough for an average to emerge, and that money is linear in usefulness — neither holds for a one-off decision or for a loss you cannot absorb.
Published an expected value calculator that reports the spread, the chance of any gain and the single most likely outcome beside the average, because a roulette number bet has an expected value of −2.7027% and its modal result is losing everything, 36 times in 37.
Reports how many repetitions it would take before the observed average resembles the expectation. On that roulette bet the spread is 216 times the expected value, which puts the figure at about 46,657 spins — the reason a casino's edge is reliable and a gambler's is not.
Carries St Petersburg, whose expected value is infinite only against an unbounded counterparty. A bank of a million caps the fair price at 10 and a billion at 15, so the value grows with the logarithm of the bankroll. The refusal to pay twenty was always correct and the model was wrong.
Reports probabilities that do not sum to 1 rather than normalising them silently, since a set summing to 1.4 is a missing or double-counted outcome and rescaling would conceal it. Also fixed a preset whose label said 'positive expected value' while computing −0.33.
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