Of the three independent events E1, E2, and E3, the probability that only E1 occurs is α, only E2 occurs is β, and only E3 occurs is γ. Let the probability p that none of events E1, E2, or E3 occurs satisfy the equations (α + β)p = αβ and (β + γ)p = 2βγ. All the given probabilities are assumed to lie in the interval (0, 1). Then Probability of occurrence of E1 / Probability of occurrence of E3 =

Let P(E1) = x, P(E2) = y, and P(E3) = z
then (1 - x)(1 - y)(1 - z) = p
x(1 - y)(1 - z) = α
(1 - x)y(1 - z) = β
(1 - x)(1 - y)z = γ
So 1 - x = p/α
So, x = α / (α + p)
Similarly y = β / (β + p) and z = γ / (γ + p)
So, P(E1) / P(E3) = α/(α + p) / γ/(γ + p) = (γ + p)α / (α + p)γ = (α + p)/(γ + p) * α/γ
Also given αβ/(α + β) = p = 2βγ/(β + γ)
⇒ β = 5 αγ/(α + 4γ)
Substituting back (α + (5 αγ/(α + 4γ)))p = α * (5 αγ/(α + 4γ))
⇒ p(α(α + 4γ) + 5 αγ) / (α + 4γ) = 5α²γ/(α + 4γ)
⇒ p(α² + 9 αγ) = 5α²γ
⇒ p(α + 9γ) = 5αγ
p = 5 αγ/(α + 9γ)
Substituting in the expression for P(E1)/P(E3)
= α/(α + 5 αγ/(α + 9γ)) / γ/(γ + 5 αγ/(α + 9γ))
= α(α + 9γ)/(α(α + 9γ) + 5 αγ) / γ(α + 9γ)/(γ(α + 9γ) + 5 αγ)
= α(α + 9γ)/ (α² + 9 αγ + 5αγ) / γ(α + 9γ)/(γ(α + 9γ) + 5 αγ)
= α(α² + 9 αγ) / γ(α² + 14 αγ) = α²(α + 9γ) / γ(α² + 14 αγ)
= α(α + 9γ)/ γ(α + 14γ)