Conditional Probability

Hi there, I'm currently trying to figure out a probability question..

A patient with a failing heart has been told that the probability of having a heart transplant is 0.38, the probability of surviving more than 12 months given that a transplant is performed is 0.85, and the probability of surviving more than 12 months given that a transplant is not performed is 0.30


determine the probability that a transplant was performed given that the patient survived more than 12 months.

I figured it's a conditional probability thing, so the thought the equation would be:

Pr (A|B) = Pr(A intersect B)/Pr (B)

with, A being the patient has a transplant and B being the patient surviving >12 months


One of the prior questions is find the probability the patient survives more than 12 months, which I calculated as 0.509, which I think plays a part in it but I can’t seem to calculate a right answer….A friend gave me his and he got .635, but I can't figure out how as it doesn't fit into the equation above...Just wondering if anyone could help at all?

Thanks

Personally, I wouldn't worry about "formulas". Here's how I would do this: imagine that there are 1000 patients. .38(1000)= 380 of them have a heart transplant, 1000- 380= 620 of them do not. Of the 380 who have a heart transplant, 85% of them, .85(380)= 323, survive more than 12 months, 380- 323= 57 of them do not. Of the 620 who do not have a heart transplant, 30%, .30(620)= 186, survive more than 12 months, 620- 186= 434 do not.

Now, "Determine the probability that a heart transplant is performed given that the patient survived more than 12 months".

Of the 323+ 186= 509, above, who survived more than 12 months, 323 had the heart transplant.