Question

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

Answer 1

0.3788

Explanation:

12 people want to teach math at a college.

6 people have a PhD.

5 people have a master's degree.

boss wants to interview 5 people for the job.

chance that all 5 of the people interviewed have a PhD.

total number of ways to select 5 applicants out of 12 is given by the combination:

combinations formula :

nCr = n! / r! * (n – r)!

C(12, 5) = 12! / (5! * 7!) = 792

number of ways to select 3 applicants with a PhD out of the 6 available is:

C(6, 3) = 6! / (3! * 3!) = 20

remaining 2 applicants can be selected from the remaining 6 applicants with a master's degree:

C(6, 2) = 6! / (2! * 4!) = 15

total number of ways to select 5 applicants with 3 having a PhD is:

20 * 15 = 300

P(3 PhDs) = 300 / 792

P(3 PhDs) = 0.3788 (rounded to four decimal places)

So, the probability of selecting five applicants for an interview where all three applicants have a PhD is approximately 0.3788

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