Expected value: the long-run average of a game, bet or decision

Expected value is the average result you would get per try if you repeated an uncertain situation many times. You find it by multiplying each possible outcome by its probability and adding the results: EV = Σ x × p. A positive expected value means you come out ahead on average; a negative one means you lose on average. I built the expected value calculator for exactly this sum, and this guide works through the examples where it matters most.

What expected value means

The idea comes from the mid-17th century, when Blaise Pascal and Pierre de Fermat corresponded about how to divide the stakes of a game that had to stop early; Christiaan Huygens published a treatise on the same principle in 1657. Expected value is not a prediction of any single result. A die cannot land on 3.5, yet 3.5 is its expected value. The number describes the long-run average, and the law of large numbers says your actual average gets closer to it as the number of tries grows.

A dice example

Imagine a game where you roll a fair die and win as many pounds as the number shown. Each face has probability 1/6, so EV = (1 + 2 + 3 + 4 + 5 + 6) ÷ 6 = £3.50. If it costs £4 to play, the expected value of playing is £3.50 − £4 = −£0.50 per game: over 100 games you would expect to be about £50 down. You can check this by rolling a few hundred times with the dice roller and watching the average settle towards 3.5. Dice probability explained: why 7 is the most common roll, and how to work out any odds covers the probabilities behind dice in more detail.

Be careful to enter net outcomes. If you pay £4 and win £6, your result is +£2, not +£6. Mixing gross prizes with a separate stake is the most common way to get an expected value wrong.

Why lotteries have negative expected value

Take a raffle with 1,000 tickets at £2 each and a single £500 prize. Each ticket has a 1 in 1,000 chance of £500, worth £0.50 on average, so the expected value of buying one is £0.50 − £2 = −£1.50. Big lotteries work the same way with smaller probabilities. A game where you pick 6 numbers from 49 has 13,983,816 possible combinations. With a £2 ticket and a £5 million jackpot, the jackpot contributes £5,000,000 ÷ 13,983,816 ≈ £0.36 to the expected value, leaving about −£1.64 per ticket before smaller prizes. Even a huge rollover rarely turns this positive, because jackpots can be shared and some of the money goes to running costs and good causes. The combinations calculator gives the number of combinations for any format, and Lottery odds explained: how the jackpot probability is calculated, and what it means explains the odds.

The house edge in roulette

A European roulette wheel has 37 pockets: 18 red, 18 black and a green zero. A £1 bet on red wins £1 with probability 18/37 and loses £1 with probability 19/37, so EV = 18/37 − 19/37 = −1/37, about −2.7p per pound. A bet on a single number pays 35 to 1: 35 × 1/37 − 1 × 36/37 is also −1/37. American wheels add a double zero, making 38 pockets, and the same bets lose 2/38, about 5.3p per pound. No betting pattern changes these numbers, because the expected value of a sum of bets is the sum of their expected values.

What expected value leaves out

Two choices with the same expected value can feel completely different. A sure £10 and a 1% chance of £1,000 both have EV £10, but the second usually pays nothing. The spread is measured by the standard deviation, which the calculator shows alongside the average. Its default example is a game with net outcomes of +100 (1%), +10 (10%) and −5 (89%): the expected value is −2.45 per play, but the standard deviation is about 11.2, so short runs can look very different from the average. A positive expected value also does not protect you from running out of money before the long run arrives.

Using it for real decisions

Expected value is how insurers price policies and how analysts compare options. If a £600 phone has a 5% chance of being lost or broken this year, the expected cost is £30, so a £50 insurance policy has negative expected value for you, though it may still be worth it if losing £600 at once would hurt. Use expected value to compare options that you will face many times, and think about the worst case for one-off decisions where a single loss matters. If your probabilities come from past counts, enter them as weights and the calculator normalises them; the probability calculator helps when you need to combine chances first.