Expected Value Calculator
Type each outcome with its probability and this expected value calculator gives the EV, variance and standard deviation, after checking that the probabilities add up to 1. Switch to betting for the EV of a bet at posted odds.
Expected value
- Expected value 100 x 0.1 + (-10) x 0.9 = 1.
- 1
- Variance The probability-weighted average of (value - EV)^2.
- 1,089
- Standard deviation The square root of the variance, in the same units as the outcomes.
- 33
- Outcomes range from
- -10 to 100
- Probabilities add up to 2 outcomes.
- 1
| Outcome | Probability | Value x probability | Probability x (value - EV)^2 |
|---|---|---|---|
| 100 | 0.1 | 10 | 980.1 |
| -10 | 0.9 | -9 | 108.9 |
How the Expected Value Calculator works
Expected value is the average result of something uncertain, weighted by how likely each result is. The expected value formula is E[X] = sum of x x p(x): multiply each outcome by its probability and add the products.
Take a game that pays 100 with probability 0.1 and costs 10 with probability 0.9. The expected value is 100 x 0.1 + (-10) x 0.9 = 10 - 9 = 1. Nobody ever gets exactly 1 from one play. It is what the results average out to over many plays.
The probabilities have to add up to 1, because together the outcomes must cover everything that can happen. A list that adds to 0.9 is missing an outcome, and the expected value it gives is off by whatever the missing outcome would contribute. This expected value calculator checks the sum before it computes anything and says how far off it is. If the numbers are weights or counts, such as 3 wins and 7 losses, tick the rescale box and each one is divided by the total.
The variance measures spread: Var[X] = sum of p(x) x (x - E[X])^2. For the example, 0.1 x (100 - 1)^2 + 0.9 x (-10 - 1)^2 = 980.1 + 108.9 = 1,089. The standard deviation is its square root, 33, in the same units as the outcomes. A fair six-sided die has an expected value of 3.5 and a standard deviation of 1.7078, and a raffle where 1 ticket in 500 wins $200 and a ticket costs $1 has an expected value of -$0.60 per ticket.
A bet has two outcomes, so its EV is the same formula with two terms: EV = p x profit - (1 - p) x stake. In decimal odds d, that simplifies to stake x (p x d - 1). The edge is p x d - 1, the expected profit per dollar staked. The break-even probability is 1 / d, the win rate at which the EV is exactly zero.
As an EV calculator for betting, it needs your estimate of the win probability. At -110, decimal 1.9091, the break-even rate is 52.38%. With a 55% estimate, a $100 bet expects 0.55 x 1.9091 - 1 = 5% of the stake, +$5.00. At +150 with a 40% estimate the EV is exactly zero, and an even-money bet on red at an American roulette wheel, 18 chances in 38, has an edge of -5.26%.
The result is only as good as that estimate. A posted price carries the bookmaker's margin, so the probability it implies is higher than a fair one, and a bet at the market's own no-vig probability has a negative EV equal to the margin.
Expected Value Calculator FAQ
What is the expected value formula?
E[X] = sum of x x p(x). Multiply each possible outcome by its probability and add the results. For 100 at 0.1 and -10 at 0.9 that is 10 - 9 = 1.
Why do the probabilities have to add up to 1?
Because the outcomes have to cover everything that can happen, and the probabilities of everything that can happen add up to 1. If they add up to less, an outcome is missing; if more, two outcomes overlap.
How do you calculate the expected value of a bet?
Multiply the win probability by the profit, then subtract the loss probability times the stake. At -110 a $100 bet wins $90.91, so with a 55% chance the EV is 0.55 x $90.91 - 0.45 x $100 = +$5.00.
What is the break-even probability?
1 divided by the decimal odds: the win rate at which the bet's EV is zero. It is 52.38% at -110 and 40% at +150. A win probability above it gives a positive EV, and one below it gives a negative EV.
What do variance and standard deviation add?
They measure how far single results land from the average. In the 100 or -10 example the EV is 1 but the standard deviation is 33, so any one play is far more likely to show -10 than anything near 1.
Does a positive EV mean a bet will win?
No. It is an average over many repetitions, it depends entirely on the probability estimate, and single results vary widely around it. If betting has stopped feeling like entertainment, the National Council on Problem Gambling runs a free, confidential helpline at 1-800-GAMBLER.