Expected Value Explained: The Maths That Shows What a Bet Is Really Worth

Expected value, often shortened to EV, is the most useful single idea in gambling maths. It tells you what a bet is worth on average if you could repeat it a very large number of times. This guide explains it in plain language with examples that apply to players in the UK, US, Ireland, Canada and Australia.

None of this is a way to win. It is a way to see what a bet costs before you place it.

What expected value means

Expected value is the average amount you win or lose per bet over a very large number of repeats. You work it out by multiplying each possible result by its probability and adding the results together.

• A positive EV bet wins money on average.
• A negative EV bet loses money on average.
• A zero EV bet breaks even on average, which is what a fair game would be.

EV is a long-run average, not a prediction for one bet. A negative EV bet can win today, and a positive EV bet can lose.

A coin example

Someone offers you £2 if a fair coin lands heads, and asks for £1 if it lands tails. EV = 0.5 x 2 + 0.5 x (-1) = +£0.50 per toss. Reverse the deal, so you pay £2 on tails and win £1 on heads, and the EV is -£0.50. Casinos and bookmakers always offer prices on the wrong side of this sum.

European roulette

A £1 single-number bet pays 35 to 1. There are 37 pockets, so the chance of winning is 1 in 37.

EV = (1/37) x 35 + (36/37) x (-1) = 0.9459 – 0.9730 = -£0.027.

You lose about 2.7 pence per £1 on average, matching the 2.70% house edge. Every standard bet on the wheel has the same EV. On an American wheel with a double zero the figure is about -5.26 pence. See RTP, house edge and payout rate for how these terms connect.

Sports betting, in two styles of odds

A tennis match is priced at 1.90 on both players. If each player truly has a 50% chance, a £10 bet has EV = 0.5 x 9 + 0.5 x (-10) = -£0.50. That is a 5% loss on average. To break even at 1.90 you would need the true chance to be at least 1 / 1.90 = 52.6%.

American odds work the same way. At -110 you stake 110 to win 100. EV = p x 100 – (1 – p) x 110. If p is 50%, EV = -5, a cost of 4.5% of the stake. The break-even chance is 110 / 210 = 52.4%. The guide to odds formats shows how to convert between them, and how sports betting works explains the margin.

Lotteries

A £2 ticket in a game that returns a little over half of sales as prizes has an EV of roughly -£0.90 to -£1. Even a big rollover rarely makes a ticket positive once you allow for shared jackpots, US tax and the very small chance of winning. Lottery EV is almost always negative.

Bonuses and EV

A bonus changes the sum, but wagering rules can reverse the apparent gain. For illustration, with invented figures, and as a hypothetical outside current Great Britain rules (the Gambling Commission capped bonus wagering at 10 times the bonus from 19 January 2026): a £50 bonus with 35 times wagering means £1,750 of bets. At a 4% edge the expected loss across that turnover is £70, against a £50 bonus. That £20 gap is only a rough turnover sum. It is not the exact cash value of the offer, which also depends on whether you would stop early, how much of your own money you deposit, expiry dates and win caps, each of which changes the result. The same sum under a 10 times requirement would be £500 of bets and an expected loss of £20. The wagering calculator on our free spend tracker and calculators page does this sum for you. Read how bonuses actually work and the worked bonus cost analysis.

Why EV does not guarantee results

With a negative EV you can still win a session. Variance means results scatter around the average. Here is an illustration with a £1 even-money bet on red in European roulette, which wins with probability 18/37:

• After 100 bets, you are ahead in roughly 35% of runs.
• After 1,000 bets, in roughly 19%.
• After 10,000 bets, in well under 1%.

The average result per bet moves towards the EV as the number of bets grows, while the gap between your total and the expected total can still widen in pounds. In this example the chance of finishing ahead keeps falling, so a loss becomes more likely. That is why casinos are happy for you to play for a long time. See variance and streaks for more on the swings.

Limits of the idea

• EV assumes you know the true probabilities. In casino games you do, but in sports betting you do not.
• It ignores the enjoyment of playing, which has value to many people.
• It says nothing about your risk tolerance. A small chance of a large prize can appeal even with negative EV, but it should be bought as entertainment.

Using EV sensibly

Use EV to compare options, such as single-zero roulette over double-zero, or a lower-margin market over a higher-margin one. Check the maths on any offer before you accept it. Do not use EV as a reason to believe you can outsmart the odds, and do not change your limits because an offer looks mathematically neat.

How this applies in each country

The maths is identical everywhere, only the labels differ. UK and Irish bookmakers usually show fractional or decimal odds, US sportsbooks show American odds, and Canadian and Australian sites commonly show decimal or American odds depending on the operator. Convert to a probability and the EV sum works in the same way.

The bottom line

Expected value is the average result of a bet over many repeats. Almost every casino game, lottery and bookmaker price has a negative EV for the player, because an edge or margin is built in. Use EV to understand what a bet costs, choose options with lower edges, and treat gambling as paid entertainment within a budget you set in advance.

Sources

• Wizard of Odds: Roulette (wizardofodds.com)
• Khan Academy: Expected value (khanacademy.org)
• Encyclopaedia Britannica: Probability theory (britannica.com)
• Internal Revenue Service: Topic 419, Gambling income and losses (irs.gov)

18+ only (21 in most US states, 19 in most Canadian provinces for casino and betting). Gambling can be addictive. Free, confidential help is available: Great Britain: England and Scotland 0808 8020 133, Wales 0808 2819 265 (free, 24 hours), and GAMSTOP (gamstop.co.uk); US: National Problem Gambling Helpline 1-800-MY-RESET (the older 1-800-522-4700 also works); Australia: Gambling Help Online 1800 858 858 and BetStop (betstop.gov.au); Ontario, Canada: ConnexOntario 1-866-531-2600; Ireland: speak to your GP or the HSE. Never gamble money you need for living costs.