Finding out whether two events are independent using conditional probability

Question

What is the conditional probably that a family of two children has two boys, given they have at last one boy? Assume that each of the possibilities BB,BG,GB,GG is equally likely, where B represents boy and G represents girl?

Where E = {BB} and F = {BB,BG,GB}.
P(E|F) = P(E&F)/P(F) = (1/4)/(3/4) = 1/3

In the above example, are the events E, that a family with two children has two boys, and F, that a family with two children has at least one boy, independent?

My approach: Two events are independent iff P(E&F) = P(E)*P(F).
So,
P(E&F) = 1/3, P(E) = 1/4, and P(F) = 3/4.

Since, P(E&F) != P(E) * P(F) the events are not independent.

Answer 1

Indeed, the probability of two boys is not independent of the probability of at least one boy. You can see this right away by the following observations:

• It is possible to not have one boy.

• If you have two boys, then it is impossible not to have at least one boy.