Can someone help me with the following problems:
1. A couple keep having children until they either have two girls or have four children. (Assume that they do not have twins, or other multiple births) and that each child they have is equally likely to be a boy or a girl independently of all other children.
i) Find the probability that they have two girls.
ii) Find the conditional probability that they have two girls given that their first child is
a boy.
iii) Find the conditional probability that their first child is a boy given that they have two
girls.
iv) Find the conditional probability that their first child is a girl given that they have two
girls.
2. Let A and B be events with P(A),P(B) > 0.
i) Show that P(Ac|B) = 1−P(A|B).
ii) Show that if P(A|B) < P(A) then P(B|A) < P(B).
iii) Illustrate both these results using question 1.