Roulette odds and mathematics

Roulette mathematics starts with three questions: which outcomes are possible, which ones win the wager, and how much each result returns. Once those are clear, probability and expected return follow. A session adds stake rules, stopping conditions and a sequence of results.

Choose the question you need answered

Keep the model visible

The ordinary examples here use fair, independent outcomes, a single-zero or double-zero wheel, fixed conventional payouts and full loss of outside wagers on zero. Each guide states changes when it needs another model. La partage and en prison require their own settlement assumptions.

Single-spin probability describes a selected group. Expected value averages every possible net result. Variance describes their spread. None of those quantities, alone, determines how one finite session ends.

Follow one chip

On standard European red, 18 outcomes win one chip and 19 lose it. Expected profit is 18/37 minus 19/37, or −1/37 chip. The wager can still win the very next spin. An expected loss is an average across possible outcomes, rather than a fractional fee removed from each round.

Repeat the same fixed one-chip wager 100 times and expected profit becomes −100/37 chips, about −2.70. Changing the stake after results changes total exposure. It does not make the next independent spin favor the newly selected wager.

Use the payout calculator for one wager, the streak calculator for a named sequence and the strategy simulator for a bounded demonstration. Read the assumptions beside a result before comparing it with another tool or table.