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})=((150-10))/(150)*((149-10))/(149)*...*((132-10))/(132)*((131-10))/(131) \\ =(140*139*...*122*121)/(150*149*...*132*131) \end{gathered}`

Then,

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

Thus, the probability is equal to 0.22778...