A committee of 4 person is to be appointed from 7 men and 3 women, what is the probability that the committee contain exactly 2 women. At least one man.

To calculate the probability of forming a committee with exactly 2 women and at least 1 man, we need to determine the total number of possible committees that can be formed and then find the number of favorable outcomes.

To find the total number of possible committees, we will use the combination formula (nCr). The formula is given by:

nCr = n! / (r!(n-r)!),

where n is the total number of people and r is the number of people in the committee.

In our case, n = 7 men + 3 women = 10, and we need to choose a committee of 4 people. So, the total number of possible committees is:

10C4 = 10! / (4! * (10-4)!) = 210.

Now, we need to find the number of favorable outcomes, which is the number of committees with exactly 2 women and at least 1 man.

To calculate this, we can break it down into two cases:

Case 1: Exactly 2 women and 2 men:
We need to choose 2 women out of 3, which can be done in 3C2 = 3 ways.
We also need to choose 2 men out of 7, which can be done in 7C2 = 21 ways.
Therefore, the number of favorable outcomes for this case is 3 * 21 = 63.

Case 2: Exactly 2 women, 1 man, and 1 more person (either man or woman):
We need to choose 2 women out of 3, which can be done in 3C2 = 3 ways.
We also need to choose 1 man out of 7, which can be done in 7C1 = 7 ways.
Finally, we can choose the remaining person, who can be either a man or a woman, from the remaining pool of 6 people. So, this can be done in 6C1 = 6 ways.
Therefore, the number of favorable outcomes for this case is 3 * 7 * 6 = 126.

Now, we can calculate the total number of favorable outcomes by summing up the results from both cases:
Total favorable outcomes = Case 1 + Case 2 = 63 + 126 = 189.

Finally, to calculate the probability, we divide the number of favorable outcomes by the total number of possible committees:
Probability = Favorable outcomes / Total outcomes = 189 / 210 ≈ 0.9 (rounded to two decimal places).

Therefore, the probability that the committee contains exactly 2 women and at least 1 man is approximately 0.9 or 90%.