Concept explainers
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
a.
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.
Here, the number of red colour numbers is 18 and the total numbers is 38. So the probability of getting red colour numbers is
The probability distribution for the possible red colour numbers is calculated as follows:
X | 1 | –1 |
|
|
|
The expected value is calculated as follows:
Thus, the expected value for red is –$0.0526.
Hence the expected value for red is –5.26 cents.
b.
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
The probability distribution for the possible red colour numbers is calculated as follows:
X | 1 | –1 |
|
|
|
The expected value is calculated as follows:
Thus, the expected value for even is –$0.0526.
Hence the expected value for even is –5.26 cents.
c.
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
The probability distribution for the possible red colour numbers is calculated as follows:
X | 35 | –1 |
|
|
|
The expected value is calculated as follows:
Thus, the expected value for 00 is –$0.0526.
Hence the expected value for 00 is –5.26 cents.
d.
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
The probability distribution for the possible red colour numbers is calculated as follows:
X | 35 | –1 |
|
|
|
The expected value is calculated as follows:
Thus, the expected value for any single number is –$0.0526.
Hence the expected value for any single number is –5.26 cents.
e.
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
The probability distribution for the possible red colour numbers is calculated as follows:
X | 17 | –1 |
|
|
|
The expected value is calculated as follows:
Thus, the expected value for 0 or 00 is –$0.0526.
Hence the expected value for 0 or 00 is –5.26 cents.
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Chapter 5 Solutions
Elementary Statistics: A Step By Step Approach
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