180Gambler > Mathematics
Expected value
Field: Mathematics. This page was last modified on 17 August 2026.
Expected value is the average outcome of a wager, weighted by how likely each outcome is. It is not a prediction of any single result, and in most games it is a number that cannot actually occur on any one wager. It is the level around which repeated results settle, and it is the quantity that separates a favourable bet from an unfavourable one.
The calculation
To compute expected value, list every outcome, multiply each by its probability, and add the results.
For a wager that returns +35 units with probability 1/37 and loses
1 unit otherwise, the sum is (1/37) x 35 + (36/37) x (-1) = -1/37,
which is -0.027 units per unit staked.
Two conventions are worth keeping straight. Expected value may be quoted as net of stake, as above, or as the total returned including the stake. The net convention is the more useful one, because a fair wager then sits exactly at zero and anything negative is a cost.
Three worked wagers
The table below works the arithmetic for three wagers with different shapes. Note that the middle wager wins far more often than the first and still returns less per unit; frequency of winning and expected value are independent properties.
| Wager | Win probability | Payout | Net expected value |
|---|---|---|---|
| Single number, single-zero wheel | 1/37 |
35 to 1 | -0.0270 |
| Red or black, single-zero wheel | 18/37 |
1 to 1 | -0.0270 |
| Single number, double-zero wheel | 1/38 |
35 to 1 | -0.0526 |
The first two rows carry the same expected value from completely different shapes of outcome. That is the practical lesson of the quantity: it collapses a whole distribution of results into one comparable figure, and in doing so it deliberately sets aside information about how bumpy the ride will be. Putting that information back is the job of variance.
Expected value is not a forecast
A wager with an expected value of -0.027 units never returns minus 0.027 units. It
returns plus thirty-five or minus one. The expected value is the long-run average of those two results in the
proportion the mechanism produces them, and it becomes a good description of an actual outcome only after the
wager has been repeated many times.
This is why a short session tells you almost nothing about the terms of a game, and why a long one tells you almost everything. The arithmetic does not change between the first wager and the ten-thousandth. Only the degree to which the observed result resembles the arithmetic changes.
Compound and conditional wagers
Wagers that resolve in stages are handled by working backwards. Compute the expected value of the final stage first, treat that number as the value of reaching the stage, and carry it back through the earlier branches. A craps point bet, a multi-leg accumulator and a bonus round that awards further spins are all handled by this method.
Where a stage offers a choice, the value of the choice is the better of the available branches, which is how optimal strategy in a game like blackjack is derived: at every decision point, the play with the higher expected value is chosen, and the value of the whole hand follows from the chain of those decisions.
What it does not measure
Expected value says nothing about how long a given sum of money survives, how large the swings will be, or how a person will feel about them. Two wagers with identical expected value can produce entirely different experiences, and a game designer chooses between them deliberately. That trade-off is the subject of game design.