If A and B are two events such that P(A)=0.7, P(B)=0.4 and P(A∩B)=0.5, where B denotes the complement of B, then P(B∣(A∪B)) is equal to:
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Step-by-Step Solution
Step 1: Find P(A ∩ B)
We know that the probability of event A can be expressed as the sum of the probability of A intersecting with B and the probability of A intersecting with the complement of B. This is because event A can be partitioned into two disjoint parts: A∩B and A∩B. Using the given values, we can find P(A∩B).
Step 2: Calculate P(A ∪ B)
The probability of the union of two events A and B is given by the formula P(A∪B)=P(A)+P(B)−P(A∩B). We have all the necessary values to compute this.
Step 3: Calculate P(A ∪ B̅)
To find P(A∪B), we first need P(B). Since P(B)=0.4, P(B)=1−P(B)=1−0.4=0.6. Now we can use the union formula for A and B.
Step 4: Calculate P(B ∩ (A ∪ B̅))
We need to find the intersection of event B with the union of A and B. Using the distributive property of set operations, B∩(A∪B)=(B∩A)∪(B∩B). Since B∩B is an empty set (null event), its probability is 0. Therefore, P(B∩(A∪B))=P(A∩B).
Step 5: Calculate P(B | (A ∪ B̅))
Finally, we use the formula for conditional probability: P(X∣Y)=P(Y)P(X∩Y). We have already calculated both the numerator and the denominator in the previous steps.