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Chapter 12: Problem 46

You work 5 evenings each week at a bookstore. Your supervisor assigns you 5 evenings at random from the 7 possibilities. What is the probability that your schedule does not include working on the weekend?

Short Answer

Expert verified
The probability that the 5 evening shifts will not include weekend is \(\frac{1}{21}\).

Step by step solution

01

Calculate Total Possible Ways

First, calculate the total number of ways to schedule 5 evening shifts from 7 days of a week. This is done using the combination formula \(C(n, r) = \frac{n!}{r!(n-r)!}\), where n is the total number of items and r is the number of items to choose. Plug in n = 7 (7 days in a week) and r = 5 (5 evening shifts), we get \(C(7, 5) = \frac{7!}{5!(7-5)!} = 21\). So, there can be 21 different possibilities to schedule 5 shifts in a week.
02

Calculate Desired Ways

Secondly, we calculate the number of ways to schedule 5 evening shifts from 5 weekdays (Monday to Friday). Here, n = 5 (number of weekdays) and r = 5 (5 evening shifts). So, plug in these values into the combination formula, we get \(C(5, 5) = \frac{5!}{5!(5-5)!} = 1\). Hence, there is only 1 way to schedule 5 evening shifts without including the weekends.
03

Calculate Probability

The probability is the ratio of the number of desired ways to the total number of possible ways. Therefore, the probability that the 5 evenings will not include the weekends is \(\frac{1}{21}\).

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