Final answer:
To show that P(A|A ∪ B) = P(A)P(b), it is important to understand the concept of mutually exclusive events...
Step-by-step explanation:
To show that P(A|A ∪ B) = P(A)P(b), it is important to understand the concept of mutually exclusive events. When two events, A and B, are mutually exclusive, it means that they cannot occur at the same time. This implies that P(A AND B) = 0.
Now, using the formula for the probability of the union of two events, we have P(A ∪ B) = P(A) + P(B) - P(A AND B). Since A and B are mutually exclusive, the probability of their intersection, P(A AND B), is zero.
Therefore, the formula simplifies to P(A ∪ B) = P(A) + P(B). Now, to find P(A|A ∪ B), we use the conditional probability formula: P(A|A ∪ B) = P(A ∪ B) / P(A). Substituting the value we obtained for P(A ∪ B) in the previous step, we get P(A|A ∪ B) = (P(A) + P(B)) / P(A).
Finally, simplifying the expression, we have P(A|A ∪ B) = (P(A) + P(B)) / P(A) = 1 + P(B)/P(A) = P(A)P(B)/P(A) = P(B).