Expectations for Roulette

The expectation value of betting times at constant odds is the same as the power of the expectation value for one bet. In roulette games the arrangement of numbers can be ignored for simple bets on individual numbers. Casinos that use European wheels with only one zero pay 35 times the bet and return the bet. Thus the expected value is (1+35)/37, that is, 36/37. American wheels have an extra zero, giving an expectation of 36/38. The plot shows what can be expected to remain after bets with expectation values 36/36, 36/37, and 36/38.