A box contains four fair dices and one crooked die with a loaded weight which makes the six-face appear on two-thirds tosses. You are asked to select one, die at random and toss it. If the crooked die is indistinguishable from the fair die and the result of your toss is a six-face; what is the probability that you tossed the crooked die?

Question: A box contains four fair dices and one crooked die with a loaded weight which makes the six-face appear on two-thirds tosses. You are asked to select one, die at random and toss it. If the crooked die is indistinguishable from the fair die and the result of your toss is a six-face; what is the probability that you tossed the crooked die?

Solution:

P (Fair dice) = 4/5
P (Crooked die) = 1/5
The probability of tossing a six face in a fair die = 1/6
The probability of tossing a six face in a crooked die = 2/3
The probability of tossing a six face when die is crooked = 1/5 x 2/3 = 2/15.
The probability of tossing a six face when die is fair = 1/5 x 1/6 = 4/30.
The probability of tossing a six face = 2/15 + 4/30 = 4/15
The posterior probability that die tossed is crooked is : 2/15 / 4/14 = 1/2 or 50%.