Consider the experiment depicted by the Venn diagram, with the sample space S containing five sample points. The sample points are assigned the following probabilities: P(E,) = .20, P(E,) = .30, P(E,) = .30, P(EJ - .lo, P(E,) = .lo.

a. Calculatc P(A), P(B), and P(AnB).

b. Suppose we know that event A has occurred, so that the reduced sample space consists of the three sample points in A-namely. El, E2, and E,. Use the formula for conditional probability to adjust the probabilities of these three sample points for the knowledge that A has occurred

[i.e., P(E,JA)].v~~~~~ that the conditional probabilities are in the same proportion to one another as the original sample point probabilities.

c. Calculate thc conditional probabilily I'(B~A)

in two ways: (1) Add the adjusted (conditional)

probabilities of the sample points in the intersection AnB, since these represent the event that B occurs given that A has occurred; (2)

Use the formula for conditional probability:

Verify that the two methods yield the same result.

d. Are events A and B independent? Why or why not?