There are 3 math clubs in the school district, with 5, 7, and 8 students respectively. Each club has two co-presidents. If I randomly select a club, and then randomly select four members of that club to give a copy of Introduction to Counting and Probability, what is the probability that two of the people who receive books are co-presidents?
+133 The probability of selecting each club is 1/3. Let M stand for normal member and let C stand for co-president. (these are distinguishable)
Case 1: first club of 5 people is selected (MMMCC)
There are 5 choose 4 = 5 ways to select the four members, and we must get a group of MMCC in our choice. There are thus 3 choose 2 = 3 ways to choose MMCC out of MMMCC. The probability here is 3/5, and the total probability of this case happening and the co-president restriction is 1/3 * 3/5 = 1/5.
Case 2: second club of 7 people is selected (MMMMMCC)
There are 7 choose 4 = 35 ways to select the four members, and we must get a group of MMCC in our choice. There are thus 5 choose 2 = 10 ways to choose MMCC out of MMMMMCC. The probability here is 10/35 = 2/7, and the total probability of this case happening and the co-president restriction is 1/3 * 2/7 = 2/21.
Case 3: third club of 8 people is selected (MMMMMMCC)
There are 8 choose 4 = 70 ways to select the four members, and we must get a group of MMCC in our choice. There are thus 6 choose 2 = 15 ways to choose MMCC out of MMMMMMCC. The probability here is 15/70 = 3/14, and the total probability of this case happening and the co-president restriction is 1/3 * 3/14 = 1/14.
1/5 + 2/21 + 1/14 = 11/30
See https://web2.0calc.com/questions/combination-probability
