What is Pr[E | F]?

Question

Suppose for this problem that Pr[E] = [ 1/12], Pr[F] = [ 1/6], and Pr[E ∩F′] = [ 0/1]. 

(1) What is Pr[E | F]?

(2) What is Pr[F | E]?

Answer 1

To solve this problem, you need to use Bayes’ Theorem for conditional probability. P(E|F) is the Probability that E occurs given F occurs. We are looking at the proportion of times E and F both occur given F occurs.

The probability of two events A and B both occurring is represented as P(A n B). P(E|F) = P(E n F)/P(F)

Note, E can occur regardless of whether F occurs or not. We can have E occur and F occurs or E occur and F not occurring when E occurs. This constitutes all of the scenarios of E occurring. P(E n F’) + (E n F) = P(E). You use similar logic for setting up the probability of F occurring. In statistics, we are using something known as a sample space. The sample space is all of the possibilities such as E n F, E n F’, E’ n F’, and E’ n F. When we look for probabilities, we want to sum the probabilities of all disjoint events that meet an outcome. Disjoint events are events that can’t happen at the same time. I can’t have E n F and E n F’ occur at the same time. Otherwise, we would have F and F’ happen simultaneously. We can’t have F occurring and F not occurring happening simultaneously for a single event.

Write these equations down and use the known probabilities given to you to solve for the unknowns.