You are given three dice D_1, D_2, and D_3: Die D_1 has 0 on two of its faces an

Question

You are given three dice D_1, D_2, and D_3: Die D_1 has 0 on two of its faces and 1 on the other four faces. Die D_2 has 0 on all six faces. Die D_3 has 1 on all six faces. You throw these three dice in a box so that they end up at uniformly random orientations. You pick a uniformly random die in the box and observe that it has 0 on its top face. Use the formal definition of conditional probability to determine the probability that the die that you picked is D_1. S = {(D_1, 0), (D_1, 1), (D_2, 0) (D_3, 1)} where, for example, (D_1, 1) is the outcome in which you observe 1 on top of die D_1. Note that this is not a uniform probability space.

Explanation / Answer

Solution:

Chances of event A which is that you pick D1 is : 1/3

Chances of event B which is that you see 0 on the top face of the die you picked is : 1/3 on D1 + 1 on D2

= 1/3 +1= 4/3

Then the probability that it is dice D1 and showing 0 will be : (1/3) / (4/3) (i.e. Pr(A|B))

= 1/4

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