Can we treat these events as independent? Problem:
The chances of someone having a genetic stigma is $1$%. Assume a couple with $2$ children. If both parent have a stigma, the chances of each child having it is $50$%. If just one parent has a genetic stigma, the chances of each child having it is $2$%.
a) What is the probability of both children having the stigma, assuming that only one parent has it?
b) What is the probability of both children having the stigma, if we know nothing about the parents?
My assumption:
I think that in both a) and b) we deal with independent events (since the birth of a child doesnt influence the chances of the next one having the stigma). So i think that for a) its $0.02 * 0.02$ and for b) its $0.01 * 0.01$
Is this assumption correct?
 A: Hint:
a) is conditional: $Pr(2C | 1P) = \frac{Pr(2C \text{ } {\cap} \text{ } 1P)}{Pr(1P)}$, where $C$ = children and $P$ = parents
b) If we nothing about the parents, we cannot tell anything about the children either. When we know nothing, we consider every option possible there is to consider... what if both parents have a stigma and both children have or one parent has a stigma and both children have or no parent has stigma and both children have.
That is, $Pr(2P \text{ } {\cap} \text{ } 2C)$, $Pr(1P \text{ } {\cap} \text{ } 2C)$, $Pr(0P \text{ } {\cap} \text{ } 2C)$
"independent events (since the birth of a child doesnt influence the chances of the next one having the stigma)." - Yes in this context, thats correct, but this is not where you are going wrong with the question.
A: Yes, in each of cases (a) and (b), the two events are independent of each other.
[You can think of this exercise as a $4$-trial (parent 1, parent 2, child 1, child 2) probability experiment, where trials one & two are pairwise independent, as are trials three & four, but all other trial pairs are dependent on each other.]
BTW, “the chances of someone having a genetic stigma is 1%” is ambiguous: is it referring to the case where neither parent has the stigma, or is it the expected probability all three cases (both, neither, exactly one parent) considered? Separately, is it also referring to each parent's probability?
