Question

A roulette wheel has the numbers 1 through 36, 0, and 00. A bet on three numbers pays 11 to 1 (that is, if you bet $1 and one of the three numbers you bet comes up, you get back your $1 plus another $11). How much do you expect to win with a $1 bet on three numbers?

Answer 1

You are expected to lose $0.05 (or win -$0.05)

Explanation:

Since the roulette wheel has the numbers 1 through 36, 0, and 00, there are 38 possible outcomes.

In this bet, you are allowed to pick 3 out of the 38 numbers. Thus, your chances of winning (P(W)) and losing (P(L)) are:

`P(W)=(3)/(38)\\P(L) = 1 - P(W)\\P(L) = (35)/(38)\\`

The expected value of the bet is given by the sum of the product of each outcome pay by its probability. Winning the bet means winning $11 while losing the bet means losing $1. The expected value is:

`EV = (11*(3)/(38)) -(1*(35)/(38))\\EV = -\$0.0525`

Therefore, with a $1 bet, you are expected to lose roughly $0.05