How this instrument works
Expected value is a probability-weighted average. For each possible outcome, multiply its value by the chance it occurs, then add those products together across every outcome. A single trial never actually lands on the expected value itself — it's not a prediction of any one result, it's the average you'd converge toward if the same uncertain situation repeated many, many times.
That long-run framing is what makes expected value useful for comparing bets, prices, and decisions that look different on the surface. A raffle ticket, an insurance policy, and a business proposal all reduce to the same shape: a set of possible outcomes, each with a value and a probability. Expected value collapses all of that into one comparable number, which is why it shows up across gambling, insurance pricing, and everyday decision-making under uncertainty.
A particularly useful reference point is zero. When the expected value of a bet is exactly zero, it's called a fair game — over many repetitions, wins and losses balance out exactly, with no edge to either side. A positive expected value favors the player over the long run; a negative one favors the house. Casino games and lotteries are built with a negative expected value for the player by design, which is exactly why they're profitable for the operator across enough plays.
- Enter the value of the first possible result into Outcome 1 value, and its likelihood into Outcome 1 probability (0-1) — for example, 100 and 0.01 for a 1% chance of winning $100.
- Enter the second possible result the same way into Outcome 2 value and Outcome 2 probability (0-1).
- Enter a third possible result into Outcome 3 value and Outcome 3 probability (0-1) — if your situation only has two outcomes, leave both of these at 0.
- Read Expected value beneath the inputs — it's the probability-weighted sum of all three outcomes.
- Keep the three probabilities summing to 1 or less; anything left over is treated as an implicit "nothing happens" outcome worth 0, and the instrument warns you if the probabilities add up to more than 1.
Worked example — a $1.50 raffle ticket
A raffle ticket gives a 1% chance of winning a $100 prize, a 5% chance of winning a $10 prize, and a 94% chance of winning nothing. Enter v1 = 100, p1 = 0.01, v2 = 10, p2 = 0.05, v3 = 0, p3 = 0.94. The instrument computes 100×0.01 + 10×0.05 + 0×0.94 = 1 + 0.5 + 0 = 1.5. Expected value reads $1.50.
That $1.50 is what the ticket is worth on average if you imagine buying thousands of them: most tickets pay nothing, a handful pay $10, and rarer still one pays $100, and those outcomes average out to $1.50 per ticket over the long run. If the ticket itself costs more than $1.50, the expected value of buying it is negative — you'd expect to lose money on average, even though any single ticket might still hit the $100 prize.
Questions
What does a positive or negative expected value actually mean?
A positive expected value means that, averaged over many repetitions, the outcome favors gaining value; a negative one means it favors losing value on average. It says nothing about any single trial — a bet with a positive expected value can still lose on any given attempt, and one with a negative expected value can still occasionally pay off big.
Does expected value predict what happens on one single trial?
No. Expected value is a long-run average across many repetitions, not a forecast for any individual outcome. In the raffle example, no single ticket actually pays out $1.50 — every ticket pays $0, $10, or $100. The $1.50 figure only emerges as the average once you look across a large number of tickets.
Why should the three probabilities sum to 1 or less?
Because probabilities represent the full set of things that could happen, and the total chance across every outcome can't exceed certainty (1, or 100%). If the three probabilities you enter sum to less than 1, the instrument treats the shortfall as an implicit fourth outcome worth 0 — like the 94% "win nothing" case in the raffle example. If they sum to more than 1, the instrument flags it as invalid.
What is a "fair game" in expected-value terms?
A fair game is one where the expected value works out to exactly 0 — over many repetitions, gains and losses balance out precisely, with no advantage to either side. A 50/50 bet of winning $50 or losing $50 has EV = 50×0.5 + (-50)×0.5 = $0, a textbook fair game: entering v1 = 50, p1 = 0.5, v2 = -50, p2 = 0.5, v3 = 0, p3 = 0 in this instrument confirms it.
Is expected value only used for gambling?
No. Insurers price policies around the expected value of claims, businesses use it to weigh uncertain projects against their costs, and it underlies decisions anywhere a choice has several possible outcomes with different values and different odds. Gambling examples are common mainly because the probabilities and payouts are unusually clean and explicit.
Can I use this instrument with just two possible outcomes?
Yes — enter your two real outcomes into Outcome 1 and Outcome 2, then set both Outcome 3 value and Outcome 3 probability (0-1) to 0. A zero-value, zero-probability outcome contributes nothing to the sum, so the result is identical to a genuine two-outcome calculation.