Can someone help with this question??

From Jon Lai on the Passed Tense team:

There is nothing wrong with your approach. You just need to solve the numerator of Pr(pass test & non survivor).

Consider that you have Pr(non survivor) = 0.4, Pr(pass) = 0.55 and Pr(pass | survivor) = 0.85. You ought to make use of that conditional probability, and it can become Pr(pass & survivor) if you multiply it by Pr(survivor). Therefore, Pr(pass & survivor) = 0.85 * (1 – 0.4) = 0.51.

So how does Pr(pass), Pr(pass & survivor), and Pr(pass & non survivor) all relate? The law of total probability: Pr(pass) = Pr(pass & survivor) + Pr(pass & non survivor). So you can solve for the numerator as

Pr(pass & non survivor) = Pr(pass) – Pr(pass & survivor) = 0.55 – 0.51 = 0.04

Divide 0.04 by 0.4, and there you have it. You’ll notice that what we just did is the same thing as the solution. That formula for Pr(pass test) simply puts all the relationships into one neat expression.

I believe you meant “not independent”. And yes, that’d be correct. You can actually tell (don’t have to assume) that independence does not exist. Pr(pass) = 0.55, while Pr(pass | survivor) = 0.85. If “passing” and “being a survivor” are independent, then – loosely speaking – it wouldn’t make a difference if the event was conditional or not, i.e. Pr( A ) = Pr( A | B ). Since the unconditional probability of passing (0.55) is not the same as the conditional event of passing (0.85), that means both events are not independent.

Now you might turn around and say: “Well that applies to “passing” and “being a survivor”; how about “passing” and “being a non survivor”?” You have the same deal when it comes to complement of events. So if events A and B are independent, then independence also exists between the pairs A & B’, A’ & B, and A’ & B’.