GATE | GATE-CS-2003 | Question 3

Let P(E) denote the probability of the event E. Given P(A) = 1, P(B) = 1/2, the values of P(A | B) and P(B | A) respectively are
(A) 1/4, 1/2
(B) 1/2, 1/14
(C) 1/2, 1
(D) 1, 1/2


Answer: (D)

Explanation: Given, P(A) = 1, P(B) = \frac{1}{2}
We need to find the conditional probability of two given events without being told about P(A\cap B). Also it is not mentioned that they are independent events.
But since P(A) is 1, it means that A covers the complete sample space.
So, P(A\cap B) = P(B) = \frac{1}{2}
 P(A|B) = \frac{P(A\cap B)}{P(B)} = \frac{1/2}{1/2} = 1\\ P(B|A) = \frac{P(A\cap B)}{P(A)} = \frac{1/2}{1} = \frac{1}{2}

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