Question

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 00, 0, or 1.
b. Which bet is​ better: a $5 bet on the number 25 or a $5 bet hat the outcome is any one of the numbers 00, 0, or 1? Why?

Answer 1

The expected value for the $5 bet that the outcome is 00, 0, or 1 is -$0.39. The $5 bet on the number 27 is the better choice.

Step-by-step explanation:

To find the expected value for the $5 bet that the outcome is 00, 0, or 1, we need to calculate the expected value for each outcome and then sum them up.

Probability of making a net profit of $30 is 5/38, so the expected value for this outcome is (5/38) * $30 = $3.95.

Probability of losing $5 is 33/38, so the expected value for this outcome is (33/38) * (-$5) = -$4.34.

Adding up the expected values, we get $3.95 - $4.34 = -$0.39.

Therefore, the expected value for the $5 bet that the outcome is 00, 0, or 1 is -$0.39.

To determine which bet is better, we compare the expected values.

The expected value for the $5 bet on the number 27 is -26, which is greater than -0.39.

Therefore, the $5 bet on the number 27 is the better choice.

Answer 2

Step-by-step explanation:

Given that:

To bet $5 that the outcome is any one of these five possibilities: 0, 00, 1, 2, 3.

Let Y represent the Amount of net profit

Then, Y= {-5, 30}

The probability distribution of Y is:

Y -5 30

P(Y=y)
`(33)/(38)`
`(5)/(38)`

a) The expected value of X is given by:

`E[Y] =\sum y P(Y=y)= 30*(5)/(38)-5*(33)/(38)`

`=-0.39`

`= -39 \ cents`

b)

On a bet of $5 on the number 25 we are expected to loose 24 cents.

While on a $5 bet that the outcome is any one of the numbers 0,00, or 1 we are expected to loose 39 cents.

Hence, $5 bet on the number 27 is better. Because the expected loss is less in this bet