Answer: Tuesday Teaser #17
Incredibly, they aren’t the same. Let’s calculate the two using conditional probability:
In the first problem, we want to find the probability that both children are boys given that at least one of them is. Using the formula for conditional probability this is equal to:
P(Both children are boys AND at least one child is a boy)/P(at least one child is a boy)
It’s clear that this is equal to P(Both children are boys)/P(at least one child is a boy)
There are four possibilities: GG, GB, BG, BB, so the probability of at least one boy is 3/4 and the probability of both being boys is 1/4.
Hence we conclude the first problem gives us an answer of 1/3.
In the second part, using the same method we get that the probability that both children are boys given that at least one of them is a boy born on a Tuesday is:
P(Both children are boys AND at least one child is a boy born on a Tuesday)/P(at least one child is a boy born on a Tuesday)
There are 296 possibilities: G(Mon)G(Mon), G(Mon)G(Tue), etc., of which 13 give us the situation where both children are boys and at least one child is a boy born on a Tuesday. Similarly, 27 give us the situation where at least one child is a boy born on a Tuesday.
So, amazingly, we end up with the probability having increased to 13/27, much closer to the intuitive answer of 1/2.
The New Scientist provides a similar explanation of how to arrive at this answer without resorting to conditional probability.
