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Chapter 4: Problem 88

A company is to hire two new employees. They have prepared a final list of eight candidates, all of whom are equally qualified. Of these eight candidates, five are women. If the company decides to select two persons randomly from these eight candidates, what is the probability that both of them are women? Draw a tree diagram for this problem.

Short Answer

Expert verified
The probability that the company will hire two women from the pool of eight candidates (from which five are women) is approximately 0.357 or 35.7%.

Step by step solution

01

Calculate Total Combinations

Calculate the total number of combinations of choosing 2 employees from the pool of 8 candidates. Using the combination formula, it is calculated as \(8C2 = \frac{8!}{2!(8-2)!} = 28\) combinations.
02

Calculate Combinations with Desired Outcome

Now, calculate the combinations with the desired outcome -- that both employees are women. There are 5 women among the 8 candidates. The number of combinations of 2 employees that can be formed from 5 women is calculated similarly as \(5C2 = \frac{5!}{2!(5-2)!} = 10\) combinations.
03

Calculate Probability

Finally, calculate the probability of the desired outcome by dividing the number of combinations with a desired outcome by the total number of combinations. The probability of the company hiring two women is therefore \(P = \frac{5C2}{8C2} = \frac{10}{28} = 0.357143\) to six decimal places.
04

Construct a Tree Diagram

A tree diagram for this problem can be created using the probability calculated. The first branch would be the probability of the first person selected being a woman (5/8), with the corresponding branch being the probability of the first person being a man (3/8). Each branch would then further branch into two parts - one for the probability of the second person being a woman, and one for the probability of the second person being a man. These probabilities would differ slightly from the original probabilities, as the pool has decreased by one. When multiplied together, the branches resulting in two women being selected should total to the calculated probability (0.357143).

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