Consider the experiment defined by the accompanying Venn diagram, with the sample space S...

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.