Question

An urn contains 5 white and 10 black marbles. A fair die is rolled and that number of marbles is randomly selected (without replacement) from the urn. What is the conditional probability that the die landed on 4 if all the marbles are white?

Answer 1

Combination formula: nCr = n!/(r! x (n-r)!)

Number of white marbles = 5

Total number of marbles = 5 + 10 = 15

P(die landed on 1 and all marbles are white) = 1/6 x 5C1/15C1

= 1/6 x 5/15

= 0.0556

P(die landed on 2 and all marbles are white) = 1/6 x 5C2/15C2

= 1/6 x 10/105

= 0.0159

P(die landed on 3 and all marbles are white) = 1/6 x 5C3/15C3

= 1/6 x 10/455

= 0.0037

P(die landed on 4 and all marbles are white) = 1/6 x 5C4/15C4

= 1/6 x 5/1365

= 0.0006

P(die landed on 5 and all marbles are white) = 1/6 x 5C5/15C5

= 1/6 x 5/15

= 0.0000

P(die landed on 1 and all marbles are white) = 0
(because there are only 5 white marbles)

P(all marbles are white) = (0.0556+0.0159+0.0037+0.0006+0.0001+0)

= 0.0758

Bayes' Theorem: P(A | B) = P(A & B) / P(B)

P(the die landed on 4 | all the marbles are white) = P(die landed on 4 and all the marbles are white)/P(all marbles are white)

= 0.0006/0.0758

= 0.008