 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/discrete-mathematics-continuous-mathematics-and-probability/8/a/i/solution

The two outcomes are not independent. Conditional on the selected die, the rolls are independent, but sharing the unknown die creates statistical dependence. Each nonfavored face has probability

(1 - 1/2) /5 = 1/10

The unconditional marginal distribution assigns the following probability to a

1

on either roll:

P (X = 1) = P (Y = 1) = \frac{1}{6} (\frac{1}{2} + 5 \cdot \frac{1}{10}) = \frac{1}{6} .

P (X = 1, Y = 1) = \frac{1}{6} [(\frac{1}{2})^{2} + 5 (\frac{1}{10})^{2}] = \frac{1}{20} \neq = \frac{1}{36} .

Thus the joint probability does not factor into the two marginals. Conditional independence given a shared latent choice does not imply unconditional independence.

 New to topics? Read the docs here!