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Mathematical model

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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 {\frac  {1}{(36+n)}}. For a betting option with p numbers that define a win, the chance of winning a bet is {\frac  {p}{(36+n)}}

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

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

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

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

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

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