Probability and Odds in Roulette

Roulette probability, odds against winning and payout odds describe different things. Start with a fair single-zero wheel: thirty-seven equally likely pockets. A straight-up bet has one winning pocket and thirty-six losing pockets. Its probability of winning is one out of thirty-seven, about two point seven percent. Its odds against winning are thirty-six to one. The standard profit payout is only thirty-five to one. For a one-dollar winning bet, that payout means thirty-five dollars profit plus the dollar stake: thirty-six dollars returned. Average the total return over all thirty-seven equally likely outcomes and subtract the stake. The expected net result is minus one over thirty-seven dollars. Coverage changes the chance of winning. A three-number street covers three out of thirty-seven pockets. A dozen covers twelve. A red bet covers eighteen. Zero is neither red nor black, so red is not a fifty-fifty bet under ordinary rules. For red, the odds against winning are nineteen to eighteen, but the ordinary profit payout is one to one. That again creates a one-over-thirty-seven house edge. Double-zero roulette has thirty-eight pockets, so recalculate the denominator. Check special zero rules separately. These are long-run expectations under stated assumptions, not promises about one spin or session.