Last quarter I taught discrete math. One component of the class was to cover some basic probability theory. On one of the homeworks we asked the following two questions about random five card poker hands:

- Given that the hand contains an ace, what is the probability that the hand contains another ace?
- Given that the hand contains the ace of diamonds, what is the probability that the hand contains another ace?

Without doing any explicit calculations, which of the above probabilities do you think will be larger?

I find this problem interesting because while I can do the calculation and understand why it comes out the way it does, I was never able to give myself a good intuitive explanation beyond comparing this problem to a much simpler case with a similar behavior. Anyone have a good inuitive explanation for why one of the above probabilities is bigger than the other? (And yes I’m avoiding telling you the answer of which one is bigger: work it out for yourself! Or see the comments were I’m sure someone will leak the correct answer.)