Roulette Expected Value
Roulette expected value is the average result of a bet if you could repeat it forever. In European roulette it is the same for every standard bet: you lose 1/37 of each stake, on average, per spin.
In plain English
Imagine placing the same $10 bet on red a million times. Some spins win $10, some lose $10. Add up every result and divide by a million: you get about −$0.27. That number is the expected value of a roulette bet. It is not what happens on any one spin; it is the long-run average per spin.
The formula for the EV of a roulette bet
p = n/37 · W = S·(36/n − 1) · n = numbers covered · S = stake
Substitute the roulette payout rule, 36/n − 1, and n cancels out. That is the whole story of the house edge in one line:
= S·(36 − n − 37 + n)/37
= −S/37
Try it: a roulette expected value calculator
Pick how many numbers the bet covers. The probability and payout change; the expected value does not.
What it looks like over time
Average result per spin for a $10 red bet, as spins accumulate. Early on, luck dominates; the running average settles toward −$0.27 as the sample grows. This is the long-run convergence that the house relies on.
Common misconceptions
Comic relief
Not what happens on one spin. The average of all of them.