Question: When to use the product rule in the context of conditional probability and when not to use?
Below are two problems that I solved using the same technique. The first problem is correct while the second is wrong.
Given a deck of cards, I have two problems that I solve exactly the same way. However, I don't ge the same answer. Could someone explain why is that so?
Problem 1: Compute P(Ace|Space)
Here's how I do it:
P(Ace|Spade) = P(Ace + Spade) / P(Spade)
So, I first compute the numerator:
P(Ace + Spade) = P(Ace) * P(Spade) = (4/52)(13/52) = 1/52
So, the answer would be:
P(Ace|Spade) = (1/52) / (13/52) = 1/13
Now consider a second problem, I have difficult with:
Problem 2: Compute P(Queen|Face Card)
I follow the same procedure as above:
P(Q|F) = P(Q + F) / P(F)
Now here's where I have the problem:
P(Q + F) = P(Q)P(F) = (4/52)(12/52)
What is wrong in the above formula as the expected answer is 4/52?
The rest of the steps are the same as above.