Back to dice. Your friend casts two dice and tells you that at least one of them is a six. What is the probability that both of them are sixes?
We use Bayes’ theorem first.
P(B6|A6) – Probability of both six, given at least one is a six.
P(A6|B6) – Probability at least one is a six given both are six = 1
P(B6) – Prior probability for both of them is six = 1/36.
P(A6|F6) – Probability at least one is a six, given the first die is a six (and the second is not) = 1
P(F6) – Prior probability of the first die is a six (and the second is not) = (1/6)x(5/6).
P(A6|S6) – Probability at least one is a six, given the second die is a six (and the first is not) = 1
P(S6) – Prior probability of the second die is a six (and the first is not) = (1/6)x(5/6).
P(A6|N6) – Probability at least one is a six given none is size = 0
P(N6) – Prior probability of none is a six = (5/6)x(5/6).
If you are not convinced, here is a pictorial explanation:
Here are all the pairs that include at least one of them as a six. There are 11 of them in total. And there is only one that is both six. So the probability is 1 in 11.
