Question

A 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 Men

Answer 1

a)

$$P(M=4, F=0)=\dfrac{\dbinom{8}{4}\dbinom{7}{0}}{\dbinom{15}{4}}=$$ $$=\dfrac{\dfrac{8!}{4!(8-4)!}(1)}{\dfrac{15!}{4!(15-4)!}}=\dfrac{70}{1365}=\dfrac{2}{39}$$

(b)

$$P(M=0, F=4)=\dfrac{\dbinom{8}{0}\dbinom{7}{4}}{\dbinom{15}{4}}=$$

$$=\dfrac{\dfrac{7!}{4!(7-4)!}(1)}{\dfrac{15!}{4!(15-4)!}}=\dfrac{35}{1365}=\dfrac{1}{39}$$

c)

$$P( F\geq1)=1-P(M=4, F=0)=1-\dfrac{2}{39}=\dfrac{37}{39}$$

d)

$$P( M\geq1)=1-P(M=0, F=4)=1-\dfrac{1}{39}=\dfrac{38}{39}$$