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)

$$P(4\ Men)=\dfrac{\dbinom{8}{4}\dbinom{7}{0}}{\dbinom{7+8}{4}}$$ $$\dbinom{8}{4}=\dfrac{8!}{4!(8-4)!}=70$$

$$\dbinom{7}{0}=1$$

$$\dbinom{8+7}{4}=\dfrac{15!}{4!(15-4)!}=1365$$

$$P(4\ Men)=\dfrac{\dbinom{8}{4}\dbinom{7}{0}}{\dbinom{7+8}{4}}=\dfrac{70}{1365}=\dfrac{2}{39}$$

b)

$$P(4\ Women)=\dfrac{\dbinom{8}{0}\dbinom{7}{4}}{\dbinom{7+8}{4}}$$ $$\dbinom{8}{0}=1$$

$$\dbinom{7}{4}=\dfrac{7!}{4!(7-4)!}=35$$

$$\dbinom{8+7}{4}=\dfrac{15!}{4!(15-4)!}=1365$$

$$P(4\ Women)=\dfrac{\dbinom{8}{0}\dbinom{7}{4}}{\dbinom{7+8}{4}}=\dfrac{35}{1365}=\dfrac{1}{39}$$

(c)

$$P(at\ least\ 1\ Women)=1-P(0\ Women)$$

$$=1-P(4\ Men)=1-\dfrac{2}{39}=\dfrac{37}{39}$$

d)

$$P(at\ least\ 1\ Men)=1-P(0\ Men)$$

$$=1-P(4\ Women)=1-\dfrac{1}{39}=\dfrac{38}{39}$$