When to use different 'given that' formula

Question

This may be a bit of a stupid question, but I genuinely don't understand.

I have the following question:

Two factories produce devices. Factory X produces 500 devices, 50 of which are defective. Factory Y produces 1000 devices, 150 of which are defective.

A device is selected at random and found to be defective; what is the probability that it came from factory X?

So I thought I could define the following events:

Event A - the device came from factory X Event B - the device is defective

And use the formula:

$P(A|B) = \frac{P(A\cap B)}{P(B)}$

However the solutioins decide to use:

$P(A|B) = \frac{P(B|A)*P(A)}{P(B)}$

When do you know which formula to use?

Answer 1

Because $$P(A\cap B) = P(B|A) \cdot P(A)$$