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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.

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.

$2.19