Let Bᵢ (i = 1, 2, 3) be three independent events in a sample space. The probability that only B₁ occur is α, only B₂ occurs is β and only B₃ occurs is γ. Let p be the probability that none of the events Bᵢ occurs and these 4 probabilities satisfy the equations ( α - 2β )p = α β and ( β - 3γ )p = 2β γ (All the probabilities are assumed to lie in the interval (0, 1)).
Then P( B₁ )/P( B₃ ) is equal to ________.

Solution

Maths sir · 4 steps

Write every probability with x, y, z; the common factors cancel and each equation turns linear.

1. Name the unknowns. Let P(B₁) = x, P(B₂) = y, P(B₃) = z. By independence, α = x(1-y)(1-z), β = (1-x)y(1-z), γ = (1-x)(1-y)z and p = (1-x)(1-y)(1-z).
2. First equation. In (α - 2β)p = αβ, both sides share the factor (1-x)(1-y)(1-z)². Cancel it, since every probability is non-zero, to get x(1-y) - 2y(1-x) = xy.

   x - xy - 2y + 2xy = xy ⇒ x = 2y

3. Second equation. In (β - 3γ)p = 2βγ, cancel (1-x)²(1-y)(1-z) to get y(1-z) - 3z(1-y) = 2yz. This gives y - 3z + 2yz = 2yz, so y = 3z.
4. So. x = 2y = 2(3z) = 6z, so P(B₁)/P(B₃) = x/z = 6.

The idea to keep. For independent events, write each probability as a product of P(Bᵢ) and 1 - P(Bᵢ). Pull out the shared factors before expanding, and the equations usually become linear.

Answer from the NTA final answer key. Solution by PrepWiser.