Boy or Girl?

This puzzle looks very simple, but is it really that simple? Hmmm, let analyze.

Problem is,
1) A (random) family has two children, and the older child is a boy. What is the probability that the younger child is a girl?
2) A (random) family has two children, and one of the children is a boy. What is the probability that the younger child is a girl?
3) A (random) family has two children, and one of the children is a boy named Jacob. What is the probability that the other child is a girl?

Solution:
1). In this problem, a random family is selected. In this sample space, there are four equally probable events:

• Both older and Younger children are Girl
• Older child is Girl and Younger child is Boy
• Old Child is Boy and Younger one is Girl.
• Both children are Boys.
  

Only two of these possible events meets the criteria specified in the question (e.g., BG, BB). Since both of the two possibilities in the new sample space {BG, BB} are equally likely, and only one of the two, BG, includes a girl, the probability that the younger child is a girl is 1/2.

2) This question is identical to question one, except instead of specifying that the older child is a boy, it is specified that one of them is a boy. Again, there are four equally probable events for a two-child family as seen in the sample space above. But sample space will now be {BB, BG, GB}. Only one out of these satisfies the case where younger child will be Girl. So, probability is 1/3.

3) Looking at the problem, it looks like sample space here is {BB,BG,GB,GG}. NO!. There could be the cases where first child is a boy and second child is also a Boy named Jacob. Based on this, we arise at following alternatives,

• Both children are Boys
• Both children are Girls
• Older child is Boy and Younger child is Jacob.
• Older child is Girl and Younger child is Jacob.
• Older child is Jacob and Younger child is Boy.
• Older child is Jacod and Younger child is Girl.

First 2 alternatives will not be included in sample space, as it doesn't have Jacob in it, and thus violating problem statement. So, our sample space will be {BJ,GJ,JB,JG}. 2 out these sample space has Girl. So, answer is 1/2??
Is this answer correct? NO!!!
Reason is, above solution doesn't take into account different frequencies of each of these answers. The likelihood of a boy being named Jacob and a boy not being named Jacob are not equal. Thus, we must replace our classical interpretation of probability with either a Frequentist or Bayesian interpretation. More details can be found at Wikipedia or CurioUser. This is one of the puzzle Adobe asks during interview.