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(E1 = .20, P(E2) = .30, P(E3) = .30, P(E4) = .10, P(E5) = .10.

a. Calculate P(A), P(B), and P(A ˆ© B)
b. Suppose we know that event A has occurred, so that the reduced sample space consists of the three sample points in A - namely, E1, E2, and E3. 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(Ei|A)]. Verify that the conditional probabilities are in the same proportion to one another as the original sample point probabilities.
c. Calculate the conditional probability P(B|A) in two ways: (1) Add the adjusted (conditional) probabilities of the sample points in the intersection A ˆ© B, as 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?

Transcribed Image Text:

PAnB) P(A) F (BIA)-