Expected Value in Roulette When playing roulette at the Venetian casino in Las Vegas, a gambler is trying to decide whether to bet \(5 on the number 27 or to bet \)5 that the outcome is any one of these five possibilities: 0, 00, 1, 2, 3. From Example 6, we know that the expected value of the \(5 bet for a single number is -26¢. For the \)5 bet that the outcome is 0, 00, 1, 2, or 3, there is a probability of 5/38 of making a net profit of \(30 and a 33/38 probability of losing \)5.

a. Find the expected value for the \(5 bet that the outcome is 0, 00, 1, 2, or 3.

b. Which bet is better: a \)5 bet on the number 27 or a $5 bet that the outcome is any one of the numbers 0, 00, 1, 2, or 3? Why?

A $5 bet is made on the five possible outcomes of 0, 00, 1, 2, or 3. The probability of winning the bet is equal to $\frac{5}{38}$, and the net profit made is equal to $30. The probability of losing the bet is equal to $\frac{33}{38}$, and the amount lost is equal to $5.

a.

The expected value of the bet is equal to the expected amount that can be won or lost.

It is computed as follows.

$$\begin{gathered} \text{Expected}\text{value}=\left(\text{Net}\text{profit}\right)\times \left(\text{Probability} \\ \text{of}\text{winning}\right)-\left(\text{Net}\text{loss}\right)\times \left(\text{Probability} \\ \text{of}\text{losing}\right) \\ =\left(30\right)\left(\frac{5}{38}\right)-\left(5\right)\left(\frac{33}{38}\right) \\ =-0.395\text{dollars} \\ =-39.5\text{cents} \end{gathered}$$

Thus, the expected value of a $5 bet on the outcomes 0, 00, 1, 2, or 3 is equal to -39.5 cents.

b.

The game that has a higher expected value is considered beneficial.

The expected value of the $5 bet on the number 27 is equal to -26 cents.

As the expected value of a $5 bet on the number 27 is greater than the expected value of a $5 bet on the 5 outcomes, the $5 bet on the number 27 is better.