Bag 1 contains 4 white and 6 black balls while another Bag 2 contains 4 white and 3 black balls. One ball is drawn at random from one of the bags and it is found to be black. Find the probability that it was drawn from Bag 1.

(a) 12/13

(b) 5/12

(c) 7/11

(d) 7/12

The question was posed to me in an interview for internship.

My question comes from Bayes Theorem topic in division Probability of Mathematics – Class 12

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The correct answer is (d) 7/12

The explanation: Let E1 = event of choosing the bag 1, E2 = event of choosing the bag 2.

Let A be event of drawing a black ball.

P(E1) = P(E2) = 1/2.

Also, P(A|E1) = P(drawing a black ball from Bag 1) = 6/10 = 3/5.

P(A|E2) = P(drawing a black ball from Bag 2) = 3/7.

By using Bayes’ theorem, the probability of drawing a black ball from bag 1 out of two bags is-:

P(E1|A) = P(E1)P(A|E1)/( P(E1)P(A│E1)+P(E2)P(A|E2))

= (1/2 × 3/5) / ((1/2 × 3/7)) + (1/2 × 3/5)) = 7/12.