Question

From a sample of 8 women and 12 men, a committee consisting of 4 different people is to be formed.

a) How many different committees are possible? Explain your answer.
b) How many different ways to form a committee with a chairperson, first vice chairperson, second vice chairperson, and secretary? Explain your answer.
c) Find the probability of having exactly 2 women on the committee. Explain your answer.
d) Find the probability of having at least 1 woman on the committee. Explain your answer.

Answer 1

a) There are 4845 different committees that can be formed. b) There are 116280 different ways to form a committee with specific positions. c) The probability of having exactly 2 women on the committee is approximately 0.3811. d) The probability of having at least 1 woman on the committee is approximately 0.8977.

Step-by-step explanation:

a) To determine the number of different committees that can be formed, we need to use combinations. With 8 women and 12 men, the total number of people to choose from is 20. We want to choose a committee of 4 people. The number of different committees is given by the combination formula: C(20, 4) = (20!)/(4!(20-4)!) = 4845.

b) To determine the number of different ways to form a committee with specific positions, we can use permutations. Since we want to choose a chairperson, a first vice chairperson, a second vice chairperson, and a secretary from the committee members, the number of ways to form the committee is given by the permutation formula: P(20, 4) = 20!/(20-4)! = 20x19x18x17 = 116280.

c) To find the probability of having exactly 2 women on the committee, we need to consider the possible combinations of women and men in the committee. The number of ways to choose 2 women from the 8 total women is C(8, 2) = (8!)/(2!(8-2)!) = 28. The number of ways to choose 2 men from the 12 total men is C(12, 2) = (12!)/(2!(12-2)!) = 66. Therefore, the total number of committees with exactly 2 women is 28 x 66 = 1848. The probability is given by (# of committees with exactly 2 women)/(total # of committees) = 1848/4845 ≈ 0.3811.

d) To find the probability of having at least 1 woman on the committee, we need to consider the total number of committees with at least 1 woman and divide it by the total number of committees. The total number of committees with at least 1 woman is the complement of the number of committees with no women. The number of committees with no women is the number of ways to choose 4 men from the 12 total men, which is C(12, 4) = (12!)/(4!(12-4)!) = 495. Therefore, the total number of committees with at least 1 woman is 4845 - 495 = 4350. The probability is given by (# of committees with at least 1 woman)/(total # of committees) = 4350/4845 ≈ 0.8977.