Question

A six-member committee is formed to plan a Christmas formal. The members
of the group are to be selected from five teachers and nine students. If the
members are randomly selected, what is the probability that it will include at
least two teachers?

Answer 1

Disclosure, its been a while since I did probability, so this may not be right, but hopefully it will help you come up with the answer.

We first want to look at the total possibilities, we have 6 open slots, to be filled from 5 teachers and 9 students. the possibility of any individual to be chosen is 6/14, the number of openings divided by the number of candidates. The makeup of teachers is 5/14. The chance for any teacher by themselves is (6/14)*(5/14). Times this by the number of open slots, which gives about 92%

Answer 2

This is sampling without replacement. It can be solved by finding the probability of 1 or 0 teachers and subtracting the result from 1.000000
\[P(0\ teachers)=\frac{9C6}{14C6}=\frac{9!\times 6!\times 8!}{6!\times 3!\times 14!}=0.027972\]
\[P(1\ teacher)=\frac{5C1\times 9C5}{14C6}=\frac{5 \times 9!\times 6!\times 8!}{5!\times 4!\times 14!}=0.209790\]
\[P(1\ or\ fewer\ teachers)=0.027972+0.209790=0.237762\]
\[P(at\ least\ 2\ teachers)=1.000000-0.237762=0.762238\]