Question

In a group there are M men and W women; m of the men and w of the women are college graduates. An individual is picked at random. Let A be the event the individual is a woman and B be the event he or she is a college graduate. Under what condition are the events A and B independent?

Answer 1

Total number of people = M+W

$P(A\cap B)=$ P(woman and college graduate) $=\frac{w}{M+W}$

$P(A)=$ P(woman) $=\frac{W}{M+W}$

$P(B)=$ P(college graduate) $=\frac{m+w}{M+W}$

In order for A and B to be independent, we should have $P(A\cap B)=P(A)P(B)$

$\frac{w}{M+W}=\frac{W}{M+W}\times \frac{m+W}{M+W}$

w(M+W) = W(m+w)

wM = WM

w = W

which is the required condition.