Include losses as negative payoffs and return of stake consistently.
Build the outcome table
List mutually exclusive outcomes so exactly one occurs on each wager. Assign each its probability and net profit or loss.
For a one-unit even-money bet winning 49% and losing 51%, EV is 0.49 × 1 + 0.51 × (−1) = −0.02 units.
From EV to house edge
House edge is the casino’s expected win divided by the stated wager denominator. A player EV of −0.02 units on a one-unit initial wager corresponds to a 2% edge.
Games with later wagers can also publish an element of risk using average total action as the denominator.
Decision EV
Strategy compares the EV of legal actions conditional on known cards and rules. The best action can still have negative value; it is simply less costly than the alternatives.
Winning probability alone is insufficient because different actions risk and return different amounts.
What EV cannot predict
A short session can finish far above or below expectation. Variance describes that spread.
Use EV to compare prices and project average cost, not to promise a specific bankroll path.
Working checklist
- List complete outcomes
- Use net payouts consistently
- Confirm probabilities sum to one
- State the denominator
- Separate expectation from prediction
This lesson explains general gambling mathematics and uses rounded examples. A game-specific figure applies only to the rules, pay table, strategy, and denominator stated with it. Last reviewed September 20, 2026.