I have a problem, I'm learning probability at the moment (I'm a programmer) and starting I have this:
(Source: Minka.) My neighbor has two children. Assuming that the gender of a child is like a coin flip, it is most likely, a priori, that my neighbor has one boy and one girl, with probability 1/2. The other possibilities—two boys or two girls—have probabilities 1/4 and 1/4.
a. Suppose I ask him whether he has any boys, and he says yes. What is the probability that one child is a girl?
b. Suppose instead that I happen to see one of his children run by, and it is a boy. What is the probability that the other child is a girl?
Now my reasoning is:
BB = 1/4 = 0.25 BG = 1/4 = 0.25 GB = 1/4 = 0.25 GG = 1/4 = 0.25
So for a., the probability of G I get it just by summing p(B,G) + p(G,B) = 0.5
And for b. p(G|B) = p(G,B)/p(B) = 0.5/0.5 = 1 that is wrong but I'm not getting why.