Multiple choice

A purse contains $2$ six-sided dice. One is a normal fair die,while the other has two $1's, $ two $ 3' s$ and two $5'$, A die is picked up and rolled. Because of some secret magnetic attration of the unfair die, there is 75% chance of picking the unfair die and a 25% chance of picking a fair die. The die is rolled and shows up the face $3$ The probability that a fair die was picked up is

  1. $1/7$
  2. $1/4$
  3. $1/6$
  4. $1/24$
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A Correct answer
Explanation

P(Fair) = 0.25, P(Unfair) = 0.75. P(3|Fair) = 1/6. P(3|Unfair) = 2/6 = 1/3. Using Bayes' Theorem: P(Fair|3) = [P(3|Fair) * P(Fair)] / [P(3|Fair) * P(Fair) + P(3|Unfair) * P(Unfair)] = [(1/6 * 1/4)] / [(1/6 * 1/4) + (1/3 * 3/4)] = (1/24) / (1/24 + 3/12) = (1/24) / (1/24 + 6/24) = (1/24) / (7/24) = 1/7.

AI explanation

Using Bayes' theorem, the probability that the fair die was picked given that a 3 is rolled is P(Fair|3) = P(3|Fair) * P(Fair) / P(3). The probability of rolling a 3 with the fair die is 1/6, and the prior probability of picking the fair die is 1/4. The total probability of rolling a 3 is (1/6)(1/4) + (1/3)(3/4) = 1/24 + 1/4 = 7/24. Therefore, the required probability is (1/24) / (7/24) = 1/7.

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