Question

Student population of a university has 30% Asian, 40% American, 20% European

and 10% African students. It is known that 40% of all Asian students, 50% of all

American students, 60% of all European students and 20% of all African students are

girls. Find the probability that a girl chosen at random from the university is an Asian.

Answer 1

Let $A_1$ denote asian student, $A_2$ denote american student, $A_3$ denote european student, $A_4$ denote african student, and $G$ denote that a student is a girl.

Given

$P(A_1)=0.3,$

$P(A_2)=0.4,$

$P(A_3)=0.2,$

$P(A_4)=0.1,$

$P(G|A_1)=0.4,$

$P(G|A_2)=0.5,$

$P(G|A_3)=0.6,$

$P(G|A_4)=0.2$

By the Law of the Total Probability

$$P(G)=P(A_1)P(G|A_1)+P(A_2)P(G|A_2)+$$ $$+P(A_3)P(G|A_3)+P(A_4)P(G|A_4)=$$ $$=0.3(0.4)+0.4(0.5)+0.2(0.6)+0.1(0.2)=0.46$$

Use the Bayes' Theorem

$$P(A_1|G)=\dfrac{P(A_1)P(G|A_1)}{P(G)}=$$ $$=\dfrac{0.3(0.4)}{0.46}={6\over 23}\approx0.260870$$