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

Step-by-step solution

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