Boy and Girl paradox

Question

I am trying to understand the boy and girl paradox. The paradox states that if a family has two children and one of them is a boy, then the probability of the other being a girl is 2/3. When you write out the set of possible outcomes { bb, bg, gb, gg } it makes a little more sense. My question is why does age/order matter? The two possible outcomes boy/girl and girl/ boy are the same right?

Answer 1

Well given that the children are different entities, you could say that order matters in this problem, hence the fact that $bg$ is not the same as $gb$. Say you have a big sister, and your aunt has an elder boy and a baby girl. Both families have a boy and a girl but the configuration is different. That is why knowing that one of them is a boy, it leaves $3$ possibilities, $2$ of which have girl, while if you know that the elder is a boy, the you only have $2$ configurations left.

Answer 2

The sample space is $\Omega := \{(B,B), (B,G), (G,B), (G,G)\}$, where the 1st element of each pair denotes the gender of the first child, and the 2nd element denotes the gender of the second child. Let us assume that each pair is equally likely, which is a reasonable assumption.

If you tell me that one of the children is a boy, then the sample space is reduced to

$$\Omega' := \{(B,B), (B,G), (G,B)\}$$

after I incorporate the information you gave me ("One of the children is a boy"). Since it is reasonable to assume that all three pairs are equally likely, the probability that the other child is a girl is given by

$$\mathbb{P} \left(\{(B,G), (G,B)\}\right) = \mathbb{P} \left(\{(B,G)\}\right) + \mathbb{P} \left(\{(G,B)\}\right) = \frac{1}{3} + \frac{1}{3} = \frac{2}{3}$$

Answer 3

It's not really the case that age or order matters. Suppose that the children were twins, and were somehow born in parallel rather than in series. Forgetting about the information that there is at least one boy, there are exactly three distinct possibilities, "two boys", "two girls", and "one of each". However, the mere fact that there are three possibilities does not imply that all three possibilities have the same probability of $1/3$.

To give another example of this, if I flip a coin it will either come up heads, come up tails, or land balanced on its edge. However, it would be a mistake to conclude that these events all have probability $1/3$.