Quick Conditional Probability Question

Question

Been going through lecture notes (ones I made myself, so who knows how accurate they are - or rather - they most certainly are not) and can't seem to understand this one example

Example: Two horses - A and B. A wins with probability 0.5, B wins with probability 0.3. What is the probability that B wins?

The calculation that follows: $$P(B|A^C)=\frac{P(B\cap A^C)}{P(A^C)}=\frac{0.3}{0.5}=0.6 $$

Now, I have no trouble with the theory, just understanding what is actually meant by the question. My questions:

1) Should the question really be "What is the probability that B wins, given that A doesn't?"

2) Is $P(B)=P(B\cap A^C)$ since $P(A\cap B)=\emptyset$ (or is that true?)

Perhaps 1) and 2) are obviously true and I just forgot to add that sentence in my notes. But I want to be sure (and be able to sleep at night :) )

Thanks for any help.

Answer 1

Both observations are correct and "obvious". But what's obvious at the time (or to an expert) isn't always obvious when you're revising. Let this be a lesson to you about high quality note-taking. (Even if it makes me a massive hypocrite)

Answer 2

The answers to both 1) and 2) seem to be 'yes'. Certainly the calculation makes sense in that context.