Roulette - Simplified mathematical model

For a roulette wheel with n green numbers and 36 other unique numbers the chance of the ball landing on a given number is ${\displaystyle {\frac {1}{(36+n)}}}$. For a betting option with p numbers that define a win, the chance of winning a bet is ${\displaystyle {\frac {p}{(36+n)}}}$

For example, betting on "red", there are 18 red numbers, ${\displaystyle p=18}$, the chance of winning is ${\displaystyle {\frac {18}{(36+n)}}}$.

The payout given by the casino for a win is based on the roulette wheel having 36 outcomes and the payout for a bet is given by ${\displaystyle {\frac {36}{p}}}$ .

For example, betting on 1-12 there are 12 numbers that define a win, ${\displaystyle p=12}$, the payout is ${\displaystyle {\frac {36}{12}}=3}$, so the better wins 3 times their bet.

The average return on a player’s bet is given by ${\displaystyle {\frac {p}{(36+n)}}\times {\frac {36}{p}}={\frac {36}{(36+n)}}}$

For ${\displaystyle n>0}$ the average return is always lower than 1 so on average a player will lose money. With 1 green number ${\displaystyle n=1}$ the average return is ${\displaystyle {\frac {36}{37}}}$, that is, after a bet the player will on average have ${\displaystyle {\frac {36}{37}}}$ of their original bet returned to them. With 2 green numbers ${\displaystyle n=2}$ the average return is ${\displaystyle {\frac {36}{38}}}$.

This shows that the expected return is independent of the choice of bet.