Chapter 5.2, Problem 19E

Roulette A roulette wheel has 38 numbers, 1 through 36, 0, and 00. One-half of the numbers from 1 through 36 are red, and the other half are black; 0 and 00 are green. A ball is rolled, and it falls into one of the 38 slots, giving a number and a color. The payoffs (winnings) for a $1 bet are as follows:

If a person bets $1, find the expected value for each.

a. Red

b. Even

c. 00

d. Any single number

e. 0 or 00

To determine

To find: The expected value for red.

Answer to Problem 19E

The expected value for red is –5.26 cents.

Explanation of Solution

Given info:

The roulette wheel has 38 numbers, 1 through 36, 0 and 00. The red colour is from 1 through 36 is 18 and the black colour numbers is 18 and 0, 00 are green.

Calculation:

The expected value of a discrete random variable is same as the mean of that random variable.

$\begin{array}{c}\text{Expected Value, }E\left(x\right)=\mu \\ ={\displaystyle \sum x}\cdot P\left(x\right)\end{array}$

Here, the number of red colour numbers is 18 and the total numbers is 38. So the probability of getting red colour numbers is $\frac{18}{38}$. Thus, the remaining $\frac{20}{38}\left(=1-\frac{18}{38}\right)$ represents the number not containing red numbers.

The probability distribution for the possible red colour numbers is calculated as follows:

X	1	–1
$P\left(x\right)$	$\frac{18}{38}$	$\frac{20}{38}$

The expected value is calculated as follows:

$\begin{array}{c}E\left(x\right)={\displaystyle \sum x}\cdot P\left(x\right)\\ =1\frac{18}{38}-1\frac{20}{38}\\ =\frac{18-20}{38}\\ =-0.0526\end{array}$

Thus, the expected value for red is –$0.0526.

Hence the expected value for red is –5.26 cents.

To determine

To find: The expected value for even.

Answer to Problem 19E

The expected value for even is –5.26 cents.

Explanation of Solution

Calculation:

Here, the number of even numbers is 18 and the total numbers is 38. So the probability of getting even numbers is $\frac{18}{38}$. Thus, the remaining $\frac{20}{38}\left(=1-\frac{18}{38}\right)$ represents the not even numbers.

The probability distribution for the possible red colour numbers is calculated as follows:

X	1	–1
$P\left(x\right)$	$\frac{18}{38}$	$\frac{20}{38}$

The expected value is calculated as follows:

$\begin{array}{c}E\left(x\right)={\displaystyle \sum x}\cdot P\left(x\right)\\ =1\frac{18}{38}-1\frac{20}{38}\\ =\frac{18-20}{38}\\ =-0.0526\end{array}$

Thus, the expected value for even is –$0.0526.

Hence the expected value for even is –5.26 cents.

To determine

To find: The expected value for 00.

Answer to Problem 19E

The expected value for 00 is –5.26 cents.

Explanation of Solution

Calculation:

Here, the number of 00 numbers is 1 and the total numbers is 38. So the probability of getting 00 numbers is $\frac{1}{38}$. Thus, the remaining $\frac{37}{38}\left(=1-\frac{1}{38}\right)$ represents probability of the numbers not getting 00 numbers.

The probability distribution for the possible red colour numbers is calculated as follows:

X	35	–1
$P\left(x\right)$	$\frac{1}{38}$	$\frac{37}{38}$

The expected value is calculated as follows:

$\begin{array}{c}E\left(x\right)={\displaystyle \sum x}\cdot P\left(x\right)\\ =35\frac{1}{38}-1\frac{37}{38}\\ =\frac{35-37}{38}\\ =-0.0526\end{array}$

Thus, the expected value for 00 is –$0.0526.

Hence the expected value for 00 is –5.26 cents.

To determine

To find: The expected value for any single number.

Answer to Problem 19E

The expected value for any single number is –5.26 cents.

Explanation of Solution

Calculation:

Here, the number of any single numbers is 1 and the total numbers is 38. So the probability of getting any single number is $\frac{1}{38}$. Thus, the remaining $\frac{37}{38}\left(=1-\frac{1}{38}\right)$ represents probability of the numbers not getting single numbers.

The probability distribution for the possible red colour numbers is calculated as follows:

X	35	–1
$P\left(x\right)$	$\frac{1}{38}$	$\frac{37}{38}$

The expected value is calculated as follows:

$\begin{array}{c}E\left(x\right)={\displaystyle \sum x}\cdot P\left(x\right)\\ =35\frac{1}{38}-1\frac{37}{38}\\ =\frac{35-37}{38}\\ =-0.0526\end{array}$

Thus, the expected value for any single number is –$0.0526.

Hence the expected value for any single number is –5.26 cents.

To determine

To find: The expected value for 0 or 00.

Answer to Problem 19E

The expected value for 0 or 00 is –5.26 cents.

Explanation of Solution

Calculation:

Here, the number of 0 or 00 is 2 and the total numbers is 38. So the probability of getting 0 or 00 is $\frac{2}{38}$. Thus, the remaining $\frac{36}{38}\left(=1-\frac{2}{38}\right)$ represents probability of the numbers not getting 0 or 00 numbers.

The probability distribution for the possible red colour numbers is calculated as follows:

X	17	–1
$P\left(x\right)$	$\frac{2}{38}$	$\frac{36}{38}$

The expected value is calculated as follows:

$\begin{array}{c}E\left(x\right)={\displaystyle \sum x}\cdot P\left(x\right)\\ =17\frac{2}{38}-1\frac{36}{38}\\ =\frac{34-36}{38}\\ =-0.0526\end{array}$

Thus, the expected value for 0 or 00 is –$0.0526.

Hence the expected value for 0 or 00 is –5.26 cents.