A and B are two independent events. The probability that both A and B occur is 1/6 and the probability that neither of them occurs is 1/3. Find the probability of the occurrence of A.
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A and B are two independent events. The probability that both A and B occur is 1/6 and the probability that neither of them occurs is 1/3. Find the probability of the occurrence of A.
A and B are two independent events. The probability that both A and B occur is 1/6 and the probability that neither of them occurs is 1/3. Find the probability of the occurrence of A.
Given that A and B are independent events
∴ P(A ∩B) = P (A). P(B) . . . . . . . . . . . . . . . . . . . . . . . . . (1)
Also given that P (A ∩ B) = 1/6 . . . . . . . . . . . . . . . . . . . . . . . . . (2)
And P(bar A ∩ B) = 1/3 …………….(3)
Also P(bar A ∩ B) = 1 – P(A ∪ B)
⇒ P(A ∩ B) = 1 – P(A) – P(B)P(A ∩ B)
⇒ 1/3 = 1 – P(A) – P(B) + 1/6
⇒ P(A) + P(B) ……..(4)
From (1) and (2) we get
Let P(A) = x and P(B) = y then eq’s (4) (5) become
x + y = 5/6, xy = 1/6
⇒ x – y = (x + y)2 – 4xy
= 25/36 – 4/6 = 1/6
∴ We get x = 1/2 and y = 1/3
Or x = 1/3 and y = 1/2
Thus P(A) = 1/2 and P(B) = 1/3 or P(A) = 1/3 and P(B) = 1/2