Question: Efron's dice. Using arbitrary numbers on the sides of dice can have surprising consequences for the game described in the Exercise. Efron's dice are sets of dice with the property that for each die there is another that beats it with larger probability when the game of Exercise is played. An example of a set of four Efron's dice is as follows:

• Die A: four 4's and two 0's

• Die B: six 3's

• Die C: four 2's and two 6's

• Die D: three 5's and three 1's

(a) Specify the events A > B, B > C, C > D, D > A.

(b) Find the probabilities that A > B, B > C, C > D, D > A.

Exercise: Consider the game where two dice, die A and die B, are rolled. We say that die A wins, and write A > B, if the outcome of rolling A is larger than that of rolling B. If both rolls result in the same number it is a tie.

(a) Find the probability of a tie.

(b) Find the probability that die A wins.