Of the three independent events E 1 , E 2 and E 3 , the probability that only E 1 occurs is α , only E…

Of the three independent events  E 1 , E 2 and E 3 , the probability that only E 1 occurs is α , only E 2 occurs is β and only  E 3 occurs is γ . Let the probability p that none of events E 1 , E 2 or E 3 occurs satisfy the equations α - 2 β p = α β and β - 3 γ p  =  2 β γ . All the given probabilities are assumed to lie in the interval 0, 1 .
Then   Probability of occurrence of E 1 Probability of occurrence of E 3 = ​

Solution

Let x, y, z be probability of E1, E2, E3 respectively
⇒ x 1 - y 1 - z = α
⇒ y 1 - x 1 - z = β
⇒ z 1 - x 1 - y = γ
⇒ 1 - x 1 - y 1 - z = P
Putting in the given relation we get x = 2y and y = 3z
⇒ x = 6 z ⇒ x z = 6

Asked in: JEE Advanced 2013 (Paper 1)

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