Expected Value (EV)

Expected value is the weighted average of every possible outcome of a bet. It is the single concept that determines whether a form of gambling is beatable, and it is indifferent to how any particular bet turns out.

Calculating it

Multiply each outcome by its probability and add them up:

EV = (probability of winning × amount won) − (probability of losing × amount lost)

Take a £10 bet at 3.00 decimal odds on something you believe is genuinely 40% likely:

• Win: 0.40 × £20 profit = +£8
• Lose: 0.60 × £10 = −£6
• EV = +£2 per £10 staked

That bet is worth placing. Not because it will win, it will lose 60% of the time, but because repeated enough times it gains £2 per attempt.

Now the same bet if the true probability is 30%:

• Win: 0.30 × £20 = +£6
• Lose: 0.70 × £10 = −£7
• EV = −£1

Same odds, same stake. The difference is entirely in the probability estimate, which is where all the difficulty lives.

The hard part is not the arithmetic

The formula is trivial. Knowing the real probability is not, and that is the whole game.

The bookmaker’s price already contains an estimate, but it is inflated by margin, the implied probability of all outcomes sums to more than 100%, with the excess being the overround. Stripping it out to recover the underlying estimate is called devigging, and it gives you the market’s honest opinion.

To have positive EV you must believe something the market does not, and be right about it more often than the margin costs you. The market is the aggregate of everyone who has bet, including people with better models and more information. Beating it is possible and rare.

Why casino games cannot be +EV

In casino games the probabilities are fixed by the rules and known exactly. There is no estimation problem, and no opportunity.

The house edge is the negative expected value, stated as a percentage of turnover. European roulette at 2.7% means every £100 staked has an EV of −£2.70, on every bet, forever. No staking system changes it, because EV is additive: a sequence of negative-EV bets is negative EV regardless of how the sizes are arranged.

The recognised exceptions all involve information or rules the operator did not intend to give you, card counting in blackjack, advantage play on mispriced promotions, matched betting against a free bet. All are narrow, most are countered quickly, and operators are free to restrict accounts that use them.

EV does not predict your night

A +EV bet loses regularly. A −EV bet wins regularly. Expected value describes the mean of a distribution, and the spread around that mean, the volatility, can be enormous.

This is why results over a small sample say almost nothing about whether a decision was good. The only way to judge a betting process in the short term is to examine the reasoning, not the outcome. Over a large enough sample the distinction disappears, but “large enough” usually means thousands of bets.