Question

When playing roulette at a​ casino, a gambler is trying to decide whether to bet ​$15 on the number 28 or to bet ​$15 that the outcome is any one of the three possibilities 00 comma 0 comma or 1. The gambler knows that the expected value of the ​$15 bet for a single number is negative $ 1.58. For the ​$15 bet that the outcome is 00 comma 0 comma or 1​, there is a probability of StartFraction 3 Over 38 EndFraction of making a net profit of ​$45 and a StartFraction 35 Over 38 EndFraction probability of losing ​$15.

a. Find the expected value for the ​$15 bet that the outcome is 00 comma 0 comma or 1.

b. Which bet is​ better: a ​$15 bet on the number 28 or a ​$15 bet that the outcome is any one of the numbers 00 comma 0 comma or 1​? ​Why?

Answer 1

We have been given that the expected value of the ​$15 bet for a single number is -$1.58.

Hence, the expected value of betting $15 on number 28 is -$1.58

a) Let Y be the net profit when bet $15 the outcome is 00,0, or 1​.

For the ​$15 bet that the outcome is 00,0, or 1​, there is a probability of 3/38 of making a net profit of ​$45 and a 35/38 probability of losing ​$15.

The probability distribution of Y is

y	P(y)
45	3/38
-15	35/38

The expected value of Y is

ans: the expected value for the ​$15 bet that the outcome is 00,0, or 1 is -$10.26

b) We can see that the expected value of $15 bet on number 28 is higher (of lower negative value), that the expected value for the ​$15 bet that the outcome is 00,0, or 1. We chose the bet with higher payoff (or lower loss)

ans: a ​$15 bet on the number 28 is better as it has a higher expected value (or has a lower expected loss).