Expected value meaning

Expected value meaning and the formula

Every bet has a set of possible outcomes, each with a probability and a result in pounds. Multiply each result by its probability and add them up: the total is the expected value. It answers the question 'on average, what does this bet do to my money?'

Take a £10 straight-up bet on a single number in European roulette. There are 37 pockets. You win £350 (35 to 1) with probability 1/37 and lose £10 with probability 36/37. EV = (1/37 × £350) − (36/37 × £10) = £9.46 − £9.73 = −£0.27. That 2.70% loss rate is the house edge of the single-zero wheel, and it applies to every standard bet on the table, as the Wizard of Odds roulette guide (opens in a new tab) also sets out.

Expected value of common casino bets

Because EV equals stake times the edge, you can estimate it for any game once you know its published RTP or house edge. Variants differ, so check the help screen for the live figure.

The pattern is clear: the lower the edge, the smaller the average cost of play, which is why low house edge games cost less per hour at the same stake. None of them has positive EV for the player.

What expected value does and doesn't tell you

EV describes the average over a huge number of bets. It doesn't predict any single session. If you stake £1 on 100 slot spins at 96% RTP, the EV is −£4, but you might finish £30 down or £50 up. How far results spread around the average is measured by standard deviation, and on high-volatility slots the spread is wide.

Over thousands of bets, the average result per bet settles closer to the EV. That's why casinos, which take millions of bets, earn close to the house edge, and why no staking pattern can turn a negative-EV game positive. The betting systems guide explains why a system like the Martingale only rearranges when you win and lose, not how much on average.

Using expected value to judge bonuses

EV is the cleanest way to compare offers. For bonus money with wagering, a quick estimate is: bonus − (wagering total × house edge).

1. Find the bonus amount, for example £20.
2. Multiply by the wagering multiple to get total wagering: £20 × 10 = £200.
3. Multiply total wagering by the house edge of the games you'll play: £200 × 4% = £8.
4. Subtract from the bonus: £20 − £8 = about £12 expected value.
5. For no-wagering spins, multiply spins × spin value × RTP: 100 × 10p × 96% = £9.60.

This is an average, not a promise; plenty of players finish with nothing from a bonus. Our bonus value guide and how to calculate wagering go further, including max bet and game weighting effects.

Expected value and UK rules

The 10x wagering cap, in force since 19 January 2026, improved the EV of bonus money because less play is needed before winnings are released. Under a 50x requirement, a £20 bonus needed £1,000 of play; at 4% edge that's an expected £40 cost, larger than the bonus itself. At 10x the expected cost is £8. Our guide to bonuses before and after the cap runs more examples.

Game fairness is covered by the Commission's remote gambling and software technical standards (opens in a new tab), which set requirements for random outcomes and game information. The RTP display rules page explains where you can find a game's return.

Why the expected value meaning matters to players

Knowing EV changes the question from 'can I win?' to 'what does this cost me on average?' A useful habit is to turn your session into an expected cost: £200 of total stakes on a 96% RTP slot costs about £8 on average. If that's an amount you're comfortable paying for the entertainment, fine; if not, lower the stakes or play less. If gambling no longer feels like paid-for entertainment, GamCare (opens in a new tab) offers free support on 0808 8020 133.

Related maths terms

• Expected value and odds: how odds and payouts create the edge.
• Gambler's fallacy: the mistaken belief that past results change future probabilities.
• Bankroll: the money you budget, which EV helps you size.

More definitions are in the glossary.

Where to play: two offers valued with EV

No-wagering spins are the simplest offers to value. Coral (18+, new players) gives 100 free spins at 10p, so £10 of spin stakes, once you deposit and bet £10 on slots; spins are on selected games, must be accepted within 48 hours and used within 7 days, and winnings have no wagering. Some payment types, including PayPal and Skrill, don't qualify. PlayOJO (18+, new UK customers, not Northern Ireland) gives 80 free spins on Big Bass Bonanza at 10p, £8 of spin stakes, for a first deposit of at least £10, with cash winnings. Multiply each by the game's RTP from its help screen for a rough EV. Terms apply; see are free spins worth it?.

18+. Gambling should be entertainment that costs money, not a way to make it. Set a deposit limit before you play: how to set deposit limits. Free, confidential help: National Gambling Helpline 0808 8020 133.