Given P(A) = 0.3, P(B) = 0.5, P(A | B) = 0.3, do the following.(a) Compute P(A and B).(b) Compute P(A or B).Step 1(a) Compute P(A and B).To compute P(A and B) means that we wish to find the probability that both A happened and B happened. We are given that P(A) = 0.3, P(B) = 0.5, and P(A | B) = 0.3.Since P(A | B) ≠ P(A), the occurrence of event B the probability that event A will occur. This implies that A and B are not independent events.So, to determine P(A and B), we can apply the general multiplication rule for events.P(A and B)  =  P(B) · P(A | B) =  (0.5) ·    =

Given P(A) = 0.4, P(B) = 0.5, P(A | B) = 0.3, do the following.(a) Compute P(A and B).(b) Compute P(A or B).

Given P(A) = 0.9 and P(B) = 0.3, do the following.(a) If A and B are independent events, compute P(A and B).(b) If P(A | B) = 0.6, compute P(A and B).Step 1(a) If A and B are independent events, compute P(A and B).To compute P(A and B) means that we wish to find the probability that both A happened and B happened.Recall that two events are independent if the occurrence or nonoccurrence of one event does not change the probability that the other event will occur.We are given that A and B are independent events, so we can use the multiplication rule for independent events. It is also given that P(A) = 0.9 and P(B) = 0.3.P(A and B)  =  P(A) · P(B) =  (0.9) ·    =

Given P(A) = 0.6 and P(B) = 0.4, do the following.(a) If A and B are independent events, compute P(A and B). (b) If P(A | B) = 0.1, compute P(A and B).

If P(A | B) = 0.6, compute P(A and B).We are given that P(A | B) = 0.6 and P(A) = 0.9. Since P(A | B) ≠ P(A), the occurrence of event B changes the probability that event A will occur. This implies that A and B are events.So, to determine P(A and B), we can apply the general multiplication rule for events. Recall that P(B) = 0.3.P(A and B)  =  P(B) · P(A | B) =  (0.3) ·    =

We found that P(A and B) = 0.15. Since the probability that both events happen at the same time is not 0, it means that events A and B have some outcomes in common. Therefore, the events A and B are not mutually exclusive.To compute P(A or B) means that we wish to find the probability that either A happened or B happened. Since the events are not mutually exclusive, we can apply the general addition rule for events. Recall that P(A) = 0.3 and P(B) = 0.5.P(A or B)  =  P(A) + P(B) − P(A and B) =  0.3 + 0.5 −  =