Puzzle – If one child is a boy, what is the probability the other is a boy?

Question:

A woman has two kids. There is one boy. What are the odds that the other child is a boy as well? How does this change if you are told the oldest child is a boy?

Solution:

We will divide the question into two parts and then solve:

i) A woman has two kids. There is one boy. What are the odds that the other child is a boy as well?

Answer: The probability of the other boy is 1/3.

Explanation:

If a woman has 2 children, then there are four possibilities:
1) The former child is a boy, and the second child is a boy (BB)
2) The former child is a boy, and the second child is a girl (BG)
3) The former child is a girl, and the second child is a boy (GB)
4) The former child is a girl, and the second child is a girl (GG)

But, already given that one child is a boy. 

So, we have three possibilities of (BB) (BG) (GB)
n(E) = both are boys = BB = 1 (number of favorable event)
n(S) = 3 (total outcomes; BB, BG, GB)

The probability of an Event = (Number of favourable outcomes) / (Total number of possible outcomes)

P(A) = n(E) / n(S)
Required probability P = n(E)/n(S) = 1/3

ii) How does this change if you are told the oldest child is a boy?

Ans. The probability of the other boy is 1/2.

Explanation:

If a woman has 2 children, then there are two possibilities:

1) The former child is a boy, and the second child is a boy (BB)
2) The former child is a boy, and the second child is a girl (BG)

So, we have two possibilities for (BB) (BG).

n(E) = both are boys = BB = 1 (number of favorable event)
n(S) = 2 (total outcomes; BB, BG,)

The probability of an Event = (Number of favourable outcomes) / (Total number of possible outcomes)

P(A) = n(E) / n(S)
Required probability P = n(E)/n(S) = 1/2.