Probability question

Question

A and B are two weak students of statistics and their chances of solving a problem in statistics correctly are 1/6 and 1/8 respectively. If the probability of their making a common error is 1/525 and they obtain the same answer, find the probability that their answer is correct.

Answer 1

A = student A got the correct answer

B = student B got the correct answer

P(A) = 1/6 and P(B) = 1/8

E₁ = Both students solve correctly

E₂ = Both students solve incorrectly

P(E₁) = (1/6)*(1/8) = 1/48

P(E₂) = (5/6)*(7/8) = 35/48

K = Both students obtain the same answer

P(K|E₁₎ = 524/525

P(K|E₂) = 1/525

Use Bayes' Theorem to find the probability that both their answers are correct.

P(E₁|K) = [P(E₁)*P(K|E₁) ÷ P(K)], where P(K) = P(E₁)*P(K|E₁) + P(E₂)*P(K|E₂).

P(E₁|K) = P(E₁)*P(K|E₁) ÷ [P(E₁)*P(K|E₁) + P(E₂)*P(K|E₂)]

= (1/48)*(524/525) ÷ [(1/48)*(524/525) + (35/48)*(1/525)]

= 524/25,200 ÷ [(524/25,200) + (35/25,200)]

= (524/25,200) ÷ (559/25,200)

= 524/559

≈ 0.9374

The probability that both of their answers are correct is about 0.9374.