Question

There are 97 men and 3 women in an organization. A committee of 5 people is chosen at random, and one of these 5 is randomly designated as chairperson. What is the probability that the committee includes all 3 women and has one of the women as chairperson?

Answer 1

The probability that the committee includes all 3 women and has one of the women as chairperson is approximately 0.0000012894.

Step-by-step explanation:

In this problem, we have a total of 100 people, consisting of 97 men and 3 women. We need to find the probability of selecting a committee of 5 people that includes all 3 women and has one of the women as the chairperson.

First, we need to calculate the total number of ways to choose a committee of 5 from a group of 100. This can be done using the combination formula:

C(100, 5) = 100! / (5! * (100-5)!) = 75,287,520 ways.

Next, we need to calculate the number of ways to choose all 3 women and one of them as the chairperson. This can be done by selecting 3 women from the group of 3 and then selecting 1 person from the remaining 97 men:

C(3, 3) * C(97, 1) = 1 * 97 = 97 ways.

Finally, we can calculate the probability by dividing the number of favorable outcomes (97) by the total number of possible outcomes (75,287,520):

Probability = 97 / 75,287,520 = 0.0000012894 (approximately).