Best Answer:
To the people who say that the question is stated incorrectly, you may need some review on common notation. Note the *huge* difference between A and A' (which can only be taken to mean NOT A in this context.) Those are very different things. In boolean algebra and probability, A' is a shorthand way of taking the opposite of A. If A=true, A'=false. In probability, if event A occurs, its inverse, event A' *did not occur.*

The probability of A', that is P(A') = 1 - P(A) (eq. 1)

Given this common assumption, the problem is indeed stated in a way that is self-consistent.

So, as you might guess, you'll need to use Bayes' theorem to solve this (because there are conditional probabilities involved.)

Bayes' theorem is usually written as:

P(A|B) = P(B|A) P(A) / P(B) (eq. 2)

You're supposed to solve for P(B) and P(A|B), which gives you two unknowns in that equation, so you'll have to work harder.

There's an extended version of Bayes' theorem that is not seen nearly as often, but can be written:

P(A|B) = ( P(B|A) P(A) ) / ( P(B|A) P(A) + P(B|A') P(A') ) (eq. 3)

This looks like the version you need. (The trick is knowing that this alternate form exists!) You're given enough information to solve for P(A|B) given this equation and the P(A') = 1 - P(A) equation (eq. 1) I gave above, so P(A') = 1 - 0.75 = 0.25. Now plug that into the extended Bayes equation (eq. 3) and you get:

P(A|B) = (0.8 * 0.75) / ( 0.8 * 0.75 + 0.6 * 0.25)

This gives P(A|B) = 0.8. This matches your expected solution above.

Now, you just need to solve for P(B). The easiest way is just to rearrange the normal Bayes equation (eq. 2 above) to solve for P(B). This gives:

P(B) = P(A) P(B|A) / P(A|B)

Plug in the numbers you have, and this becomes:

P(B) = 0.75 * 0.8 / 0.8

Which clearly cancels to:

P(B) = 0.75

Which also matches your expected answer. See the links below for a source of the alternate Bayes equation.

Source(s):
http://www.stat.yale.edu/Courses/1997-98/101/condprob.htm