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.


$$ P(A, B) = P(A)\,P(B) $$

Only if $A$ and $B$ are independent. Card suits are independent of card types, being Queen is obviously related to being face card. For dependent events the rule does not apply, if it did, it would mean that the events are independent by definition.


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