Expected value explained for betting

In one sentence

Expected value (EV) is the average amount a bet wins or loses per go if you could repeat it endlessly at the same odds and the same true chance.

How it works

A single bet either wins or loses, which tells you very little. EV asks a better question: across hundreds of identical bets, what would you average? If that average is positive the bet is worth taking, even though any one of them can lose.

You need two things: the odds on offer and your own estimate of the true probability. The odds tell you what you win and lose, your probability tells you how often each happens. Multiply and add, and you have EV.

The hard part is never the sum. It is getting a probability that is better than the market's. An EV figure is only as good as the probability you feed it.

The maths

EV = p \times (O - 1) \times S - (1 - p) \times S

\text{ROI} = p \times O - 1

EV: expected profit in £ per bet.
p: your estimate of the true win probability.
O: the decimal odds taken.
S: your stake in £.
ROI: expected return per £1 staked, as a fraction.

In plain English: what you win times how often you win, minus what you lose times how often you lose.

Worked betting example

Football, Match Odds. You back the away team in Brighton v Wolves at 3.00 with a £10 stake. Your model says it wins 38% of the time. The market's implied chance is 1 ÷ 3.00 = 33.3%.

If it wins: profit = £10 × (3.00 − 1) = £20.
If it loses: loss = £10.
EV = 0.38 × £20 − 0.62 × £10 = £7.60 − £6.20 = £1.40 per bet.
ROI = 0.38 × 3.00 − 1 = 0.14, so 14% per £1 staked.

Now place it on Betfair with 2% commission on winnings:

Profit if it wins: £20 × 0.98 = £19.60.
EV = 0.38 × £19.60 − £6.20 = £7.448 − £6.20 = £1.248, about £1.25.

Commission took about a tenth of the edge.

Note what EV does not say: this bet still loses 62% of the time. Ten bets like this could easily all lose.

Where it's good

Deciding whether to take any single price, pre-match or in-play.
Comparing two opportunities with different odds on a like-for-like basis.
Setting a minimum price you will accept for a selection.
Checking whether a trading strategy's average trade is positive after costs.
Feeding staking plans such as Kelly, which size bets in proportion to edge.

Limitations and pitfalls

Garbage in, garbage out. A 2% error in your probability can turn a positive EV into a negative one, especially at short odds.
Most people overestimate their edge. If your model regularly shows 15% EV against Betfair prices, suspect the model before the market.
EV ignores variance. Two bets with the same EV can have very different swings, and a positive-EV strategy can still go bust with poor staking.
EV is a long-run average. Over 50 bets luck dominates, and you will not see your EV in the results.

How to build it

Plain Python, pandas or a spreadsheet; numpy for vectorising across thousands of bets.
Data: your model probabilities, the odds actually matched (not the odds you saw), commission rate and stakes.
Practical tip: log predicted EV for every bet at the time you place it, then compare total predicted EV with actual profit after a few hundred bets. A big gap means your probabilities are off.

Implied probability: the market's probability to compare yours against.
Commission-adjusted EV: EV with Betfair commission built in properly.
Law of large numbers: why EV only shows up over many bets.
Closing line value: a faster way to check whether your EV estimates are real.
Kelly criterion: turning EV into a stake size.