# Consider the experiment defined by the accompanying Venn

Consider the experiment defined by the accompanying Venn diagram, with the sample space S containing five sample points. The sample points are assigned the following probabilities: P(E1) = .1, P(E2) = .1, P(E3) = .2, P(E4) = .5, P(E5) = .1.

a. Calculate P (A), P (B), and P(A ∩B).

b. Suppose we know that event A has occurred, so the reduced sample space consists of the three sample points in A: E1, E2, and E3. Use the formula for conditional probability to determine the probabilities of these three sample points given that A has occurred. Verify that the conditional probabilities are in the same ratio to one another as the original sample point probabilities and that they sum to 1.

c. Calculate the conditional probability P(B |A) in two ways: First, sum P(E2 |A) and P(E3 |A), since these sample points represent the event that B occurs given that A has occurred. Second, use the formula for conditional probability:

Verify that the two methods yield the same result.