Here is the original question.
Of three possible events, event A is independent of the other two, and events B and C are mutually exclusive. The probabilities that the individual events A, B, and C will occur are 0.5, 0.3, and 0.2, respectively. What is the probability that both event A and event C will occur?
The answer to this question is:
Start with the “mutually exclusive” events, as this is the most restrictive statement. If event B happens, event C cannot happen. Likewise, if event C happens, event B cannot happen. It is possible that neither event B or C will happen, but they can’t both happen.
Consider the possibilities, starting with whether event B happens.
If event B occurs, event C cannot occur, so there is no way for both event A and event C to happen. (i.e. Probability of both A and C is zero if B occurs.) If event B does not occur, event C might happen, as might event A.
Thus, the probability that both event A and event C will occur is the probability that B will NOT happen, A will happen, and C will happen. {NOTE: "and" means multiply probabilities, "or" would mean add probabilities.}
P(A and C) = P(not B) × P(A) × P(C) P(A and C) = [1 – 0.3] × 0.5 × 0.2 = 0.7 × 0.5 × 0.2 = 0.07 = 7%
The correct answer is 7%
Source: http://www.manhattanprep.com/gre/ChallengeProblems/LastWeek/
My question is:
1) P(A,C) = 0.07 in this question. However, this is not P(A)*P(C)=0.10, despite A and C are independent. Why does the rule P(A,C) = P(A)*P(C) fail even though A and C are independent? Is there a certain restraint that applies to this rule?
2) Event B and C are not independent. However, the problem states that P(A,C,~B) = P(A)*P(C)*P(~B). I thought this was possible only if C and ~B are independent. Can you please explain if this is valid?
Your help is greatly appreciated. Have a wonderful day.