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?
Tell me more
×
Mathematics Stack Exchange is a question and answer site for
people studying math at any level and professionals in related fields. It's 100% free, no registration required.
|
|
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 $1/3$. |
|||
|
|
|
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 $$\mathbb{P} \left(\{(B,G), (G,B)\}\right) = \mathbb{P} \left(\{(B,G)\}\right) + \mathbb{P} \left(\{(G,B)\}\right) = \frac{1}{3} + \frac{1}{3} = \frac{2}{3}$$ |
|||||||||||
|
|
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. |
|||
|
|
|
Try it with coins: First, flip a nickel and a dime. There quite clearly are four possibilities: (N=H, D=H), (N=H, D=T), (N=T, D=H), and (N=T, D=T). Each, also quite clearly, has a probability of 1/4; so the probability of two heads is 1/4, the probability of two tails is 1/4, and the probability of one of each is 1/2. Then repeat the same experiment with two absolutely identical quarters. There now seem to be three combinations we can discern: (Both H), (Both T), and (one H and one T). But do you really think the odds change to 1/3 for each combination just because the coins are now identical? I certainly hope not. Even though you can't see the difference between the coins, they still are different coins, and the ways that the different combinations can occur still depend on that difference. When we count possibilities for two-child families, age-order matters only because we have to acknowledge that the two children are different from each other. Even if we aren't given any information that distinguishes them, that difference still matters. But as for the probability paradox you mentioned, the answer depends on how you "know" that one is a boy. If it is because you asked if the family had any boys, then every BB, BG, or GB family would answer "yes" and the answer is 1/3. But if you learned that fact incidentally, then you could have learned a BG of GB family has a girl. You can only count half of them, and the answer is 1/2. |
|||
|
|
