You are given three dice D1, D2, and D3:2

• Die D1 has 0 on two of its faces and 1 on the other four faces.
• Die D2 has 0 on all six faces.
• Die D3 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 D1.

Hint: You want to determine Pr(A | B), where A is the event that you pick D1 and B is the event that you see a 0 on the top face of the die that you picked. There are different ways to define the sample space S. One way is to takeS = {(D1, 0),(D1, 1),(D2, 0),(D3, 1)},where, for example, (D1, 1) is the outcome in which you observe 1 on top of die D1. Note that this is not a uniform probability space.