What condition can make this event independent? A population consists of F females and M males;the population includes f female smokers and m male smokers. If A is the event that the individual is female and B is the event he or she is a smoker , find  the  condition on f, m, F and M so that A and B are independent events?
The answer is f/F=m/M
I dont know how to get that answer. Please help.
Thank you.
 A: The events $A$ and $B$ are independent precisely  if 
$$\Pr(A\cap B)=\Pr(A)\Pr(B).\tag{1}$$ 
If $F+M\ne 0$, we have
$$\Pr(A)=\frac{F}{F+M},$$
and
$$\Pr(B)=\frac{f+m}{F+M}.$$
Also,
$$\Pr(A\cap B)=\frac{f}{F+M}.$$
Substituting in (1) we get after some minor cancellation that $A$ and $B$ are independent precisely if
$$f=\frac{F(f+m)}{F+M}$$
or equivalently if
$$f(F+M)=Ff +mF.$$
Cancel the $fF$. We get $fM=mF$. 
Reamrks: $1.$ The suggested answer $\frac{f}{F}=\frac{m}{M}$ is almost but not quite equivalent to $fM=mF$.  For example, if there are no males, then $\frac{m}{M}$ does not make sense, but we still have independence. In our calculation we assumed $F+M\ne 0$. If $F+M=0$, we have independence. So in fact the given condition is not equivalent to independence. But the non-equivalence is for trivial cases the author presumably did not consider, namely $F=0$ or $M=0$.
$2.$ We did a formal calculations, since sometimes one has independence even when "intuition" might suggest we do not. But here the intuition is reasonably clear. The events $A$ and $B$ are independent if $\Pr(B|A)=\Pr(B)$. That means that given the information that a person is female gives no information about whether or not she is a smoker. That will be the case if the proportion of smokers among females is the same as it is among males, that is, if $\frac{f}{F}=\frac{m}{M}$. 
