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Three independent events: given the chances that only one occurs and that none occurs, find P(B1)/P(B3)

The same question, asked 2 times: 2021 Paper 1, 2021 Paper 7. Every wording below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 1

    82
    Let Bi(i=1,2,3)B_{i}(i= 1,2,3) be three independent events in a sample space. The probability that only B1B_{1} occur is α\alpha, only B2B_{2} occurs is β\beta and only B3B_{3} occurs is γ\gamma. Let pp be the probability that none of the events BiB_{i} occurs and these 4 probabilities satisfy the equations (α−2β)p=αβ(\alpha- 2\beta)p=\alpha\beta and (β−3γ)p=2βγ(\beta- 3\gamma)p= 2\beta\gamma (All the probabilities are assumed to lie in the interval (0,1))(0,1)). Then P(B1)P(B3)\frac{P\left( B_{1} \right)}{P\left( B_{3} \right)} is equal to

  2. Question 2

    2021 Paper 7

    84
    Let there be three independent events E1,E2E_{1},E_{2} and E3E_{3}. The probability that only E1E_{1} occurs is α\alpha, only E2E_{2} occurs is β\beta and only E3E_{3} occurs is γ\gamma. Let ' pp ' denote the probability of none of events occurs that satisfies the equations (α−2β)p=αβ(\alpha - 2\beta)p = \alpha\beta and (β−3γ)p=2βγ(\beta - 3\gamma)p = 2\beta\gamma. All the given probabilities are assumed to lie in the interval (0,1)(0,1). Then,  Probability of occurrence of E1 Probability of occurrence of E3\frac{\text{~Probability of occurrence of~}E_{1}}{\text{~Probability of occurrence of~}E_{3}} is equal to