The first step in dealing with any probability problem is to define your events. You'll see why in a bit. Let's just deal with the first question.
S=(The person selected is a smoker)
We are told that P(S)=1/4
C=(The person selected has cancer)
We are told that P(C)=1/300
We also have an event "the person selected is a smoker GIVEN THAT they have cancer." It's in slightly different words in the problem, but that's what it's saying. Why did I reword it? So that we can easily write the event as
(S|C). This is read "S given C"
And we're told that P(S|C)=3/4.
Now, what are we trying to find? In part a), it's P(C|S). Think about why this is if it's not clear. In part b), it's P(S U C). Again, make sure you understand why this is the probability we're interested in. If you don't, let me know and we can talk about that.
OK, so why all the letters instead of words? Well, your formulas in your book are all in letters, so now we just need to find relationships between the things we know, P(S), P(C), and P(S|C), and the things we don't know, P(C|S) and P(S U C).
I'll leave this part to you--look carefully through your book and you should be able to find a formula for each one. The reason I want you to do it this way is because it's good practice to turn everything into symbolic language--you'll realize that most of these problems are basically the same in structure; only the numbers change.