Let there be 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 p ' ' denote the probability of none of events occurs that satisfies 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 is equal to ________.

Question

Let there be 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 p ' ' denote the probability of none of events occurs that satisfies 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 is equal to ________.,

MARKS App · Accepted Answer

Let P E 1 = P 1 ; P E 2 = P 2 ; P E 3 = P 3 P E 1 ∩ E ¯ 2 ∩ E ¯ 3 = α = P 1 1 - P 2 1 - P 3 … 1 P E ¯ 1 ∩ E 2 ∩ E ¯ 3 = β = 1 - P 1 P 2 1 - P 3 … 2 P E ¯ 1 ∩ E ¯ 2 ∩ E 3 = γ = 1 - P 1 1 - P 2 P 3 … 3 P E ¯ 1 ∩ E ¯ 2 ∩ E ¯ 3 = p = 1 - P 1 1 - P 2 1 - P 3 … 4 Given that, ( α - 2 β ) p = α β ⇒ P 1 1 - P 2 1 - P 3 - 2 1 - P 1 P 2 1 - P 3 p = P 1 P 2 1 - P 1 1 - P 2 1 - P 3 2 ⇒ P 1 1 - P 2 - 2 1 - P 1 P 2 = P 1 P 2 ⇒ P 1 - P 1 P 2 - 2 P 2 + 2 P 1 P 2 = P 1 P 2 ⇒ P 1 = 2 P 2 … 1 and similarly, ( β - 3 γ ) P = 2 β γ P 2 = 3 P 3 … 2 So, P 1 = 6 P 3 ⇒ P 1 P 3 = 6

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