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Chapter 6: Problem 19

Six new employees, two of whom are married to each other, are to be assigned six desks that are lined up in a row. If the assignment of employces to desks is made randomly, what is the probability that the married couple will have nonadjacent desks? (Hint: First find the probability that the couple will have adjacent desks, and then subtract this number from 1.)

Short Answer

Expert verified
The probability that the married couple will have nonadjacent desks when the employees are assigned randomly to the desks is \(\frac{2}{3}\).

Step by step solution

01

Total arrangements of employees

There are 6 employees that can be arranged in 6! ways. So, the total number of desk arrangements for the employees is 6! (factorial).
02

Consider the married couple as a single unit

Now, let's consider the married couple as one unit. We will place the married couple together and treat them as a single unit. In this scenario, we have 5 units (4 single employees and 1 married couple) that can be arranged in 5! ways.
03

Calculate the number of arrangements for the married couple

Since the married couple can be arranged in two different ways (husband followed by wife or the wife followed by husband), we must multiply the arrangements calculated in Step 2 by 2. Thus, there are 2 * 5! arrangements in which the married couple has adjacent desks.
04

Calculate probabilities

Now, we will calculate the probability of the married couple having adjacent desks and having nonadjacent desks. 1. The probability of the married couple having adjacent desks is the ratio of the number of arrangements in which they have adjacent desks to the total number of arrangements for the employees. This is (2 * 5!) / 6!. 2. The probability of the married couple having nonadjacent desks is the complement of the probability of them having adjacent desks. Therefore, \(P(\text{Nonadjacent desks}) = 1 - P(\text{Adjacent desks})\).
05

Calculate the final probability

We have all the necessary probabilities, so let's calculate the probability of the married couple having nonadjacent desks. \(P(\text{Nonadjacent desks}) = 1 - P(\text{Adjacent desks}) = 1 - \frac{2 * 5!}{6!} = 1 - \frac{2 * 120}{720}\) \(P(\text{Nonadjacent desks}) = 1 - \frac{240}{720} = 1 - \frac{1}{3} = \frac{2}{3}\) Hence, the probability that the married couple will have nonadjacent desks is \(\frac{2}{3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factorial calculations
In discrete mathematics, factorial calculations are vital when determining arrangements or permutations. A factorial, denoted by an exclamation mark '!', is the product of all positive integers up to a given number. For instance, the factorial of 6 is written as \(6!\), which equals \(6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\).

Factorials help us calculate the total number of ways to arrange a set number of items. In the exercise, six employees can be arranged in \(6!\) different ways. This calculation is crucial when figuring out possible desk assignments in a row.

Understanding how to use factorials helps solve various combinatorial problems, making it easier to comprehend complex scenarios involving order and arrangements.
Complementary probability
Complementary probability is a powerful concept in probability theory that deals with finding the probability of the opposite of a known event. It simplifies calculations by focusing on what is 'not wanted'. If you know the probability of an event, you can easily find its complement by subtracting it from 1.

In the given problem, instead of directly calculating the probability that the married couple will have nonadjacent desks, it's simpler to find the probability that they have adjacent desks and use complementary probability. This is done by finding that probability and then subtracting it from 1.

This method is particularly useful in reducing complex probability problems to simpler ones and is crucial for efficiently solving problems like these.
Combinatorial arrangements
Combinatorial arrangements involve the study of different ways to organize or arrange a particular set of items. In the exercise, a married couple is treated as one single unit to simplify the problem of arranging six individuals.

By treating them as a unit, we reduce the problem to arranging five units instead of six. These five units can be arranged in \(5!\) different ways. However, since the couple itself can be arranged within their unit in two different ways (e.g., one can sit before the other), we multiply \(5!\) by 2. This approach simplifies calculations and gives a clear understanding of different arrangement strategies.

Combinatorial thinking aids in breaking down complex arrangement scenarios, making it easier to handle various combinations and permutations in a structured way.

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