If this post wasn't already at -6 I'd downvote it, for failing to provide the correct form of the puzzles, which has the second person ask the first about this additional info. Otherwise you need to consider the reason the person is providing you this information.

For the "Suppose I tell you that I have 2 children and one of them is a boy" we have the following priors:

P(two girls) = 25%

P(two boys) = 25%

P(one of each) = 50%

Consider however the following possibilities.

Scenario A) The guy is a sexist who would loudly proclaim the existence of sons, and avoid discussing the existence of daughter. Then the probabilities he has two boys is 0%. If he had two boys, he would have told you both of them are boys. By mentioning only one, he unintentionally revealed he had only one. Putting differently P(mentions only one boy|two boys) ~= 0

Scenario B) The guy thought to himself "I'll randomly pick a child and mention its gender". Then with no gender bias we have

P(mentions a boy) = 50%

P(mentions a girl) = 50%

and we go to:

P(mentions one boy|two girls)= 0%

P(mentions one boy|two boys) = 100%

P(mentions one boy|one of each) = 50%

Therefore P(two boys|mentions one boy) = P(mentions one boy|two boys) * P(two boys)/P(mentions one boy) = 100% * 25% / 50% = 1/2

So if the guy randomly picked a child who's gender he picked to mention, possibility of two boys is 50% given the info he provided.