Expected value in gambling

What expected value in gambling means

Expected value, or EV, answers a simple question: if you made this exact bet a huge number of times, what would you win or lose per bet on average? The answer is rarely a result you could actually get on one bet. A £10 roulette bet never loses exactly 27p; it wins or loses a whole amount. But across thousands of identical bets, the average settles on that figure.

Casino Math by Hannum and Cabot (opens in a new tab) defines EV as the sum of each possible net payoff multiplied by its probability. That one sentence powers nearly all of casino maths. The house edge is simply EV written as a percentage of the stake, and RTP is 100% plus that negative percentage. Browse the casino maths hub for the full set of related ideas.

The expected value formula and how to use it

Work out the EV of any bet in five steps. You need only the payout table and the probability of each outcome, both of which UK casinos must make available before you bet.

1. List every outcome the bet can have: win, lose, push, or several winning levels.
2. Write the net result of each in pounds. A win at 35 to 1 on £10 is +£350; a loss is −£10; a push is £0.
3. Find each outcome's probability by counting: pockets on a wheel, combinations of dice or card frequencies.
4. Multiply each net result by its probability.
5. Add the products. The total is the EV per bet; divide by the stake for the house edge.

Check your arithmetic: the probabilities must add up to exactly 1. If they do not, an outcome is missing. Our page on casino probability basics covers the counting step in detail.

Expected value on roulette bets

A £10 straight-up bet on a single-zero wheel wins £350 with probability 1/37 and loses £10 with probability 36/37. Its EV is (350 − 360) ÷ 37 = −£0.27, or −2.70% of the stake. A £10 bet on red wins £10 with probability 18/37 and loses £10 with probability 19/37, for an EV of −£10/37, again about −27p. Very different bets, identical price.

The French La Partage rule adds a third outcome to even-money bets: on zero you lose only half. The EV becomes 18/37 × £10 − 1/37 × £5 − 18/37 × £10 = −£0.135, halving the cost. On a double-zero wheel, red wins 18 times in 38 and loses 20, giving −£0.53 per £10. Our pages on European roulette and American roulette compare the two wheels.

EV of common £10 bets compared

Baccarat is a useful example of a three-outcome bet. On Banker, 45.86% × £9.50 = £4.357 and 44.62% × £10 = £4.462. The tie, at 9.52%, contributes nothing because the stake is returned. The difference, about −£0.106, is the EV. The Banker bet page goes through the commission in more detail, and the Sic Bo house edge page covers every dice bet.

Using EV to compare decisions

EV is more than a price tag. In games with choices, it is how strategy is built: at every decision, the better play is the one with the less negative EV. Basic strategy in blackjack is simply a table of the highest-EV action for each hand against each dealer card, which is why it reduces the house edge but cannot remove it. See our blackjack basic strategy guide.

Three Card Poker shows how much the choice can matter. Using the most common Ante Bonus table, playing Queen-6-4 or better and folding the rest gives a 3.37% house edge. Copying the dealer's queen-high rule gives 3.45%. Playing every hand regardless gives 7.65%.

Rule choices work the same way. A blackjack table paying 6 to 5 instead of 3 to 2 costs an extra 1.39% of every bet, which is about −14p per £10 hand or −£13.90 per 100 hands, before any decisions are made. Our 3:2 against 6:5 comparison and the Three Card Poker strategy guide apply the numbers.

Working out the EV of a whole session

Because EV adds up, the EV of a session is just the EV per bet times the number of bets, or equivalently total staked × house edge. Five hundred £1 spins on a 96% RTP slot means £500 staked and an EV of −£20. Three hours of £10 single-zero roulette at 60 spins an hour is £1,800 staked and an EV of −£48.60, the same sum casinos use to estimate a player's theoretical loss.

The trap is turnover. Winnings that get re-staked count as new stakes, so the EV is calculated on the total wagered, not on the money you started with. A £50 deposit can easily support several hundred pounds of turnover on a slot, and the edge applies to all of it. That is why our cost per hour guide bases every estimate on stakes and pace rather than deposits.

EV tells you the centre of the range, not the result. Over a short session your outcome is driven far more by variance than by the edge.

Side bets through an expected value lens

Side bets are where EV analysis earns its keep, because their payouts look generous while their probabilities are hidden. The baccarat Tie paying 8 to 1 has an EV of about −£1.44 per £10, more than thirteen times the Banker bet's cost. The Caribbean Stud progressive side bet averages a 26.46% house edge according to Wizard of Odds, an EV of about −£2.65 per £10. Some side bets are much milder, such as the Ultimate Texas Hold'em Trips bet at about 0.90% on its best table, so it pays to look each one up. Our answer to are casino side bets worth it compares the common ones.

Expected value of free spins and bonuses

EV also helps you see what a promotion is worth on average, and what it costs to use. One hundred spins at 10p each represent £10 of stakes. On a game with a 96% RTP, the average return from those spins would be about £9.60, though the actual result can be anything from nothing to much more. Any qualifying bet carries its own EV: £10 staked on a 96% slot has an EV of −£0.40.

For deposit bonuses with wagering, the house edge applies to every pound wagered. A £50 bonus with 10x wagering requires £500 of bets; on a 96% RTP slot the EV of that wagering is −£20. Maximum bet rules, game restrictions and time limits all shape how this works in practice. Our guides to what a casino bonus is worth and wagering requirements go through the terms.

Expected value myths

• 'Combining bets can create a positive EV.' EVs add. If every component bet is negative, so is the combination. A spread such as the James Bond layout costs the same 2.70% as each of its parts.
• 'Changing stake sizes improves EV.' Each bet's EV is its stake times the edge, whatever you bet before. Wizard of Odds' betting systems analysis (opens in a new tab) quotes the mathematician Billingsley: 'No betting system can convert a subfair game into a profitable enterprise.'
• 'A big jackpot makes a slot worth playing.' The jackpot is already built into the published RTP. A bigger prize usually means it is rarer.
• 'Negative EV means I will lose every session.' It means you lose on average. Many individual sessions finish ahead, which is part of the entertainment.

UK rules that help you calculate EV

Every EV calculation needs payouts and probabilities, and in Great Britain the Gambling Commission's RTS 3 rules on game information (opens in a new tab) require online casinos to make the rules, the payouts and the house edge, RTP or win probability available before you bet. Online slots carry stake limits of £5 per spin for adults aged 25 and over and £2 for 18 to 24-year-olds, which caps how quickly a negative EV can accumulate on a slot. Our UK RTP information page explains what to look for.

Expected value and our partner offers

The welcome offers shown here are slot-focused and every pound wagered under them carries the game's negative EV. Casumo: 18+, opt in for 50% up to £100 on a first deposit of £20 or more, 10x wagering within 7 days, £5 maximum bet, game, play and payment restrictions apply. 10bet: 18+, new UK customers get 100% up to £25 on a first deposit within 7 days, wagered 10x on slots within 3 days, maximum bonus winnings £1,000, Skrill and Neteller deposits excluded. Coral: 18+, deposit and bet £10 on slots for 100 free spins at 10p on selected games, accept within 48 hours, valid 7 days, no wagering on winnings, some payment methods excluded.

Terms apply. Table and live games are often excluded or weighted lower, so check the terms before using bonus funds on them.

18+. Every casino game has a built-in house edge, so over time you should expect to lose what you stake. Set a budget before you play and stop when it’s gone: safer gambling tools and support. Free, confidential help: National Gambling Helpline 0808 8020 133.