Conditional probability involving two conditions

Question

I would like to know if there is a way to calculate

P(A|(B and C))

without having to use P(A and B and C)? I know P(A|B) and P(A|C), and I am assuming that B and C are independent. Thank you in advance.

Answer 1

Suppose everyone is either a brown-eyed male, a blue-eyed male, a brown-eyed female, or a blue-eyed male, that 1/4 of the population falls into each group, and that the members of exactly one group are good at skiing.

Let $B$ be the event that I am male, and let $C$ be the event that I am brown-eyed. $B$ and $C$ are certainly independent.

Let $A$ be the event that I'm good at skiing.

Now can you compute, based on this information, the value of $P(A|B\hbox{ and } C)$? In other words, if you know I am a brown-eyed male, can you determine the probability that I'm good at skiing?

Answer: It's either zero or one, depending on whether brown-eyed males are or are not the one group that's good at skiing. In other words, it's either zero or one depending on the value of $P(A\hbox{ and } B \hbox{ and } C)$. But without knowing that value, you're stuck.