Question

Twelve people apply for a teaching position in mathematics at a local college. Six have a PhD and six have a master’s degree. If the department chairperson selects three applicants at random for an interview, find the probability that all three have a PhD

Answer 1

Using the hypergeometic distribution, it is found that there is a 0.0175 = 1.75% probability that all three have a PhD.

The applicants are chosen without replacement, hence the hypergeometric distribution is used to solve this question.

What is the hypergeometric distribution formula?

The formula is:

$P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}$

$C_{n,x} = \frac{n!}{x!(n-x)!}$

The parameters are:

• x is the number of successes.

• N is the size of the population.

• n is the size of the sample.

• k is the total number of desired outcomes.

In this problem, the values of the parameters are as follows: N = 20, k = 6, n = 3.

The probability that all three have a PhD is P(X = 3), hence:

$P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}$

$P(X = 3) = h(3,20,3,6) = \frac{C_{6,3}C_{14,0}}{C_{20,3}} = 0.0175$

0.0175 = 1.75% probability that all three have a PhD.

More can be learned about the hypergeometic distribution at https://brainly.com/question/24826394