Hi Math1059
Are explanations for some of these things not given in your course notes or textbook?
I can try to help with some conceptual understanding
Let's say A and B are events that can occur. You can think of them as subsets of an abstract mathematical space of all possible outcomes (what is known as a "set"). In the branch of math known as set theory, the \(\displaystyle \cap \) symbol is used to mean the intersection (in a Venn diagram sense) of two sets, i.e. the things (in this case the circumstances) that lie within both sets.
Therefore the notation \(\displaystyle A \cap B \) refers to circumstances in which both A occurs AND B occurs. \(\displaystyle P(A \cap B) \) means the probability of event A occurring AND event B also occurring.
A conditional probability is a probability of some event occurring subject to some other given condition. For example the notation P(A|B) means the probability of event A occurring given that we know it is true that event B has occurred. (So event B is the condition here, and we want to know the probability of A occurring given that condition. That's why it's called the conditional probability of A occurring).
Similarly P(B|A) is the probability of B occurring given that we know for sure that event A has occurred. (This is the conditional probability of B given A).
It turns out from probability theory that the probability of A AND B is given by the following product of probabilities
\(\displaystyle P(A \cap B) = P(A|B)P(B) \) [Equation 1]
This makes sense. Suppose that B has a 10% chance of occurring: P(B) = 0.1. And suppose the conditional probability of A occurring given B has occurred is 50%: P(A|B) = 0.5. So the probability of A occuring AND B occurring, is the probability of B occurring (10% of the time) times the probability of A occurring given B, which happens in half of those cases, i.e. 5% of the time.
\(\displaystyle P(A \cap B) = P(A|B)P(B) = 0.5\times 0.1 = 0.05 \)
EDIT (added all of this stuff below after initially posting)
Similarly, we can write the probability of B AND A in terms of a conditional probability:
\(\displaystyle P(B \cap A) = P(B|A)P(A) \) [Equation 2]
This leads to a very interesting and important result in probability theory. Basically, A AND B occurring is exactly the same outcome as B AND A occurring. In both cases, these two events have both occurred. Since they are the same outcome, their probabilities are equal
\(\displaystyle P(A \cap B) = P(B \cap A) \)
We can substitute in the expressions for \(\displaystyle P(A \cap B) \) and \(\displaystyle P(B \cap A) \) that use conditional probabilities, from Equations 1 and 2 above, to obtain:
\(\displaystyle P(A | B)P(B) = P(B | A)P(A) \)
Or
\(\displaystyle P(A | B) = \frac{P(B | A)P(A)}{P(B)} \)
This result, called Bayes' Theorem (named after the dude who came up with it) has very wide-reaching implications in statistics and probability theory. I think this is all the information you need to solve the problem. The only other notation I see in your original problem statement is \(\displaystyle B^\prime\), which means NOT B. So \(\displaystyle P(B^\prime)\) is the probability that B does NOT occur. What is the relationship between P(B) and P(B')? Hint: With 100% certainty (P=1) SOMETHING always occurs, either B or NOT B. So the probabilities P(B) and P(B') must sum up to...?
