Final answer:
a. 715 committees can be formed. b. 280 committees can be formed with exactly 2 men and 2 women. c. 630 committees can be formed with at least 2 men. d. The probability of forming a committee with all men is approximately 0.098.
Step-by-step explanation:
a. To find the number of committees that can be formed, we need to choose 4 people out of the 13 total individuals in the group. This can be done using the combination formula. So, the number of committees that can be formed is C(13, 4) = 715.
b. To find the number of committees that have exactly 2 men and 2 women, we need to choose 2 men out of 8 and 2 women out of 5. This can be done using the combination formula. So, the number of committees that can be formed is C(8, 2) * C(5, 2) = 28 * 10 = 280.
c. To find the number of committees that have at least 2 men, we can calculate the number of committees with exactly 2 men, exactly 3 men, and exactly 4 men and then add them. The number of committees with exactly 2 men is calculated in part b. The number of committees with exactly 3 men is C(8, 3) * C(5, 1) = 56 * 5 = 280, and the number of committees with exactly 4 men is C(8, 4) = 70. Therefore, the total number of committees with at least 2 men is 280 + 280 + 70 = 630.
d. To find the probability that the four person committee will be all men, we need to consider the number of ways to choose 4 men from the total group of 8 men. This can be calculated using the combination formula. So, the probability is C(8, 4) / C(13, 4) = 70 / 715 ≈ 0.098.