Let B i i = 1 , 2 , 3 be three independent events in a sample space. The probability that only B 1 occur is α , only B 2 occurs is β and only B 3 occurs is γ . Let p be the probability that none of th

Question

Let B i i = 1 , 2 , 3 be three independent events in a sample space. The probability that only B 1 occur is α , only B 2 occurs is β and only B 3 occurs is γ . Let p be the probability that none of the events B i occurs and these 4 probabilities satisfy the equations α - 2 β p = α β and β - 3 γ p = 2 β γ (All the probabilities are assumed to lie in the interval 0 , 1 Then P B 1 P B 3 is equal to______.

TestHub · Accepted Answer

Let P B 1 = p 1 , P B 2 = p 2 , P B 3 = p 3 given that p 1 1 - p 2 1 - p 3 = α … i p 2 1 - p 1 1 - p 3 = β … ii p 3 1 - p 1 1 - p 2 = γ … iii and 1 - p 1 1 - p 2 1 - p 3 = p … iv ⇒ p 1 1 - p 1 = α p , p 2 1 - p 2 = β p and p 3 1 - p 3 = γ p Also β = α p α + 2 p = 3 γ p p - 2 γ ⇒ α p - 2 α γ = 3 α γ + 6 p γ ⇒ α p - 6 p γ = 5 α γ ⇒ p 1 1 - p 1 - 6 p 3 1 - p 3 = 5 p 1 p 3 1 - p 1 1 - p 3 ⇒ p 1 - 6 p 3 = 0 ⇒ p 1 p 3 = 6

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