Hi everyone. I am struggling a bit with this one example question I am doing.
Problem:
There is a group of 12 people, 6 men and 6 women. A committee is to be formed consisting of 5 members from this group. Find the probability that Anne (one of the females) and her friend Billy (one of the males) are on the same committee.
My attempt:
I know that the total number of ways of selecting a committee is C(12,5).
I know that after Anne and Billy are chosen, there are 3 remaining places in the committee, and there are C(10,3) ways of doing this. But I am not sure what to do after this. At first, I thought this would work but it doesn't:
P = (C(6,1)*C(6,1)*C(10,3))/(C(12,5))
10C3 committees with Anne and Billy out of 12C5 possible committees
what fraction (probability) of possible committees are they on?
Hi Scott. Hmm, I am still unsure what you mean by that.
I guess the probability of choosing Anne, in particular, is 1/12, and then the probability of choosing Billy after that becomes 1/11. So maybe 1/12 * 1/11 * (10C3)/(12C5) ? I am quite confused.
Oh I simply divide those two results I mentioned, but I am not sure why that works. Because can't, Anne and Billy, be placed in the committee in different ways? 10C3 / 12C5 makes it seem like the ways in which Billy and Anne can be arranged in the committee doesn't matter.
You're on the right track! To solve this problem, you need to determine the total number of favorable outcomes (Anne and Billy being on the same committee) divided by the total number of possible outcomes (forming a committee of 5 members from the group of 12 people).
To find the total number of ways to choose Anne and Billy for the committee, you correctly used C(6,1) * C(6,1) = 6 * 6 = 36. This is because you need to select 1 woman out of 6 (C(6,1) ways) and 1 man out of 6 (also C(6,1) ways).
Now, you need to determine the number of ways to choose the remaining 3 people from the remaining 10 individuals (after Anne and Billy are chosen). The number of ways to select the remaining 3 committee members is given by C(10,3) = 120.
Finally, you can calculate the probability by dividing the number of favorable outcomes by the total number of outcomes:
P = (number of favorable outcomes)/(total number of outcomes) = (36 * 120) / C(12,5).
Therefore, the correct expression for the probability is:
P = (C(6,1) * C(6,1) * C(10,3))/(C(12,5)).
I hope this explanation helps!