# Expected Value and Law of Large Numbers in Roulette

American roulette has 38 equally sized spaces two spaces are green and have numbers 0 and 00 on them and the rest numbered from 1 to 36. Half of the non-green spaces are red and half are black.

The three-number bet, also a "row bet", is a bet on the three numbers in a vertical row on the roulette table. For the three-number bet, find the expected value of each $1 and round your answer to three decimal places. Also, apply the Law of Large Numbers to interpret the result and round your answer to one decimal place.

The twelve-number bet on twelve numbers, also a "column" bet, is on any one of the three columns of twelve numbers running horizontally along the table. For the twelve-number bet, find the expected value of each $1 and round your answer to three decimal places. Also, apply the Law of Large Numbers to interpret the result and round your answer to one decimal place.

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#### Solution Preview

The row bet pays 11:1. So, the probability of not winning is 35/38 (losing a dollar) and of winning $11 3/38. The expected value to four decimal places is -0.0526 or $0.9474.

From the Law of Large Numbers, over time regardless of short term wins or losses, the theoretical probability will be arrived at. ...

#### Solution Summary

In this solution we find the expected value of each $1 three-number bet and interpret with the Law of Large Numbers.