91Ó°ÊÓ

A discrete probability distribution, the hypergeometric distribution. It's utilized when you want to know how likely it is to get a certain number of successes from a given sample size without replacement

Random variable in simple terms generally refers to variables whose values are unknown, therefore, in this case X is the number of people on the committee who are not technically proficient.

Make the list of values that you want to use X may take on.

As we can see there is an upper bound for the situation at hand so,

$X=0,1,2,3,4,5,6,7,8,9,10$

The random variable X refers to the number of trials before the first success. Each trial is independent of others and has similar probability of success.

The distribution of X is$X~B(8,20,10)$

The expected number of instructors on the committee who are not technically proficient is :

$E\left(X\right)=\frac{n×r}{r+b}$

where, $n=10,r=8,b=20$

$$\begin{gathered} E\left(X\right)=\frac{10×8}{8+20} \\ E\left(X\right)=2.8571 \end{gathered}$$

To find the probability of at least five means it can be five or less than five.

The probability can also be shown as,

The probability that at least five on the committee are not technically proficient is:

$$\begin{gathered} P(Xâ‰\yen 5)=1-P(X≤4) \\ P(Xâ‰\yen 5)=1-0.92279 \\ P(Xâ‰\yen 5)=0.07721 \end{gathered}$$

To find the probability of at most three, that means the probability should be more than three.

The probability that at most three on the committee are not technically proficient is:

Using an excel sheet we can find the the required probability,