### 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.