Question

An University wishes to hire 4 faculty members for its newly established department. They have to select from a group of 8 Male and 7 Female applicants, all of whom are equally qualified to fill in the positions. If University selects 4 at random, what is the probability that ; a) All four will be Men b) All four will be Women c) Atleast one will be a Women d) Atleast one will be a Me

Answer 1

The University wishes to hire 4 faculty members from a group of 8 Male and 7 Female applicants i.e. total 8 + 7 = 15 applicants.

This can be done in $15 \choose4$ = 1365 number of ways, which is the number of total outcomes.

We have, the total number of ways of choosing a Men and b Women from a group of A Men and B Women is: ${A\choose a}{B\choose b}$

a) All four will be Men

Here, A = 8, B = 7, a = 4, b = 0

$\therefore$ The number of ways of choosing 4 Men = ${8\choose 4}{7\choose 0}$ = 70

$\therefore$ The probability that all four will be Men = (no of favorable cases)/(no of total outcomes)

= 70/1365 = 0.0513 (rounded to 4 decimal places)

Answer: The probability that all four will be Men is 0.0513.

b) All four will be Women

Here, A = 8, B = 7, a = 0, b = 4

$\therefore$ The number of ways of choosing 4 Women = ${8\choose 0}{7\choose 4}$ = 35

$\therefore$ The probability that all four will be Women = (no of favorable cases)/(no of total outcomes)

= 35/1365 = 0.0256 (rounded to 4 decimal places)

Answer: The probability that all four will be Women is 0.0256.

c) At least one will be a Women

The probability that at least one will be a Women

= 1 - The probability that there will be no Women [since, P(A^c) = 1 - P(A)]

= 1 - The probability that all four will be Men

= 1 - 0.0513 [by using answer a)]

= 0.9487

Answer: The probability that at least one will be a Women is 0.9487.

d) At least one will be a Men

The probability that at least one will be a Men

= 1 - The probability that there will be no Men [since, P(A^c) = 1 - P(A)]

= 1 - The probability that all four will be Women

= 1 - 0.0256 [by using answer b)]

= 0.9744

Answer: The probability that at least one will be a Men is 0.9744.