If A and B are two events such that P(A) = 0.7, P(B) = 0.4 and P(A ∩ B̅) = 0.5, where B̅ denotes the complement of B , then P(B | (A ∪ B̅)) is equal to

Solution

Maths sir · 4 steps

Intersect B with the condition first: the B-bar part drops out and only A∩B is left on top.

1. Find A and B. A splits into the part inside B and the part outside it. So P(A ∩ B) = P(A) - P(A ∩ B̅) = 0.7 - 0.5 = 0.2.
2. Bottom of fraction. P(B̅) = 1 - 0.4 = 0.6. Then P(A ∪ B̅) = P(A) + P(B̅) - P(A ∩ B̅) = 0.7 + 0.6 - 0.5 = 0.8.
3. Top of fraction. B ∩ (A ∪ B̅) = (B ∩ A) ∪ (B ∩ B̅). Since B ∩ B̅ is empty, this is A ∩ B, with probability 0.2.

   P(B | A ∪ B̅) = P(A ∩ B)/P(A ∪ B̅) = 0.2/0.8

4. So. The answer is 1/4, option (c).

The idea to keep. In P(X | Y), always work out X ∩ Y first using the distributive law. Pieces like B ∩ B̅ vanish and the numerator becomes easy.

Answer from the NTA final answer key. Solution by PrepWiser.