Question

At our high school you have 10 friends. If there are 150 total students, and a committeeis to be made from 20 students. Find the probability the committee has at least one of your friends.I want answer and explanation.

Answer 1

Notice that

$\begin{gathered} P(at\text{ least 1 friend})+P(zero\text{ friends})=1 \\ \Rightarrow P(at\text{ least 1 friend})=1-P(zero\text{ friends}) \end{gathered}$

Thus, notice that the event of picking one student to be a member of the committee is without replacement (once a student is a member of the committee, they cannot be elected again).

Therefore, the probability that none of your friends becomes a member of the committee is

$\begin{gathered} P(zero\text{ friends})=\frac{(150-10)}{150}*\frac{(149-10)}{149}*...*\frac{(132-10)}{132}*\frac{(131-10)}{131} \\ =\frac{140*139*...*122*121}{150*149*...*132*131} \end{gathered}$

Then,

$\Rightarrow P(zero\text{ friends})=\frac{140!}{150!}*\frac{130!}{120!}=0.22778...$

Thus, the probability is equal to 0.22778...