Question

A cupboard A has 4 red carpets and 4 blue carpets and a cupboard B has 3 red carpets and 5 blue carpets. A carpet is selected from a cupboard and the carpet is chosen from the selected cupboard such that each carpet in the cupboard is equally likely to be chosen. Cupboards A and B can be selected in \(\frac{1}{5}\) and \(\frac{3}{5}\) ways respectively. Given that a carpet selected in the above process is a blue carpet, find the probability that it came from the cupboard B.

(a) \(\frac{2}{5}\)

(b) \(\frac{15}{19}\)

(c) \(\frac{31}{73}\)

(d) \(\frac{4}{9}\)

I have been asked this question in an interview for internship.

This key question is from Discrete Probability in chapter Discrete Probability of Discrete Mathematics

Answer 1

Right answer is (b) \(\frac{15}{19}\)

The explanation is: The probability of selecting a blue carpet = \(\frac{1}{5} * \frac{4}{8} + \frac{3}{5} * \frac{5}{8} = \frac{4}{40} + \frac{15}{40} = \frac{19}{40}\). Probability of selecting a blue carpet from cupboard, P(B) = \(\frac{3}{5} * \frac{5}{8} = \frac{15}{40}\). Given that a carpet selected in the above process is a blue carpet, the probability that it came from the cupboard A is = \(\frac{\frac{15}{40}}{\frac{19}{40}} = \frac{15}{19}\).