Brain Puzzle Solution

I know, I know...You've all been waiting impatiently for the resolution of the latest Brain Puzzle. The Brain Puzzle stated:

The Smith's have two children; at least one of the children is a girl. What is the probability that both children are girls?

Another plausible answer could be 25%. This is due to the fact that if there is a 50% chance of a child being a girl, then the probability of two children being girls would be 25%. The problem with this is that since we already know that one of the children is a girl, then we don't really need to multiply the probabilities.

It might be tempting to say that the answer is 50%. Why would we say this? Simply because we know that the probability of a baby being a girl is ~50%. So, if one of the children is a girl, and there is a 50% probability of the second child being a girl, then the answer seems to be 50%.

Here's why this is a paradox: the answer is not 50%.

Let's think of it this way: If you have two children, there are three possible combinations that could arise: two girls, two boys, or one boy and one girl. The sum of the probabilities of these three outcomes is 100%. (Let's ignore the possibility of strange genderless babies). Their respective probabilities boil down like this:

• Girl (50%) x Girl (50%) = 25% probability
• Boy (50%) x Boy (50%) = 25% probability
• Boy (50%) x Girl (50%) = 50% probability

Why is the Boy/Girl option weighed more heavily? One reason is the simple fact that we know the sum of the probabilities must equal 100%. Tim addressed another reason in his comment: there are two permutations which would allow for the Boy/Girl combination, whereas there is only one permutation each that would allow for the Girl/Girl or Boy/Boy outcomes. Think of it like rolling dice. The reason that 7 is the most commonly rolled number is because there are more ways of rolling 7 than there are of rolling any other number. There is only one way to roll a 12 (6 + 6) but you can roll a seven with 1 + 6, 2 + 5, 3 + 4, 4 + 3, 5 + 2, 6 + 1. So, you are six times more likely to roll a 7 than a 12. With the kids there are two ways of getting Boy/Girl: by having a boy first and a girl second, or by having a girl first and a boy second.

With that said, the answer to the brain puzzle is 33%.

Boy/Boy = 25% (eliminated since the puzzle states that there is at least one girl)
Boy/Girl = 50%
Girl/Girl = 25%

So, the possible outcomes are either Girl/Boy, or Girl/Girl. Since we know that the probabilities of having a Boy and a Girl is twice as great as the probability of having two boys, we can do the simple arithmetic:
25 / 75 = x / 100
x = 33
The solution to the problem is 33%.

Congratulations to Tim who gets one million virtual yen to spend in my e-blog store.Follow up question: what is the probability that the people in this picture wish it hadn't ended up on the internet?

Outraged? Post your diatribe in the comments section.