for C is therefore With a similar calculation, the probability of D winning over A is Best overall die The four dice have unequal probabilities of beating a die chosen at random from the remaining three: As proven above, die A beats B two-thirds of the time but beats D only one-third of the time. The probability of die A beating C is (A must roll 4 and C must roll 2). So the likelihood of A beating any other randomly selected die is: Similarly, die B beats C two-thirds of the time but beats A only one-third of the time. The probability of die B beating D is (only when D rolls 1). So the likelihood of B beating any other randomly selected die is: Die C beats D two-thirds of the time but beats B only one-third of the time. The probability of die C beating A is . So the likelihood of C beating any other randomly selected die is: Finally, die D beats A two-thirds of the time but beats C only one-third of the time. The probability of die D beating B is (only when D rolls 5). So the likelihood of D beating any other randomly selected die is: Therefore, the best overall die is C with a probability of winning of 0.5185. C also rolls the highest average number in absolute terms, . (A's average is , while B's and D's are both 3.) Variants with equal averages Note that Efron's dice have different average rolls: the average roll of A is , while B and D each average , and C averages . The intransitive property depends on which faces are larger or smaller, but does not depend on the absolute magnitude of the faces. Hence one can find variants of Efron's dice where the odds of winning are unchanged, but all the dice have the same average roll. For example, A: 7, 7, 7, 7, 1, 1 B: 5, 5, 5, 5, 5, 5 C: 9, 9, 3, 3, 3, 3 D: 8, 8, 8, 2, 2, 2 These variant dice are useful, e.g., to introduce students to different ways of comparing random variables (and how comparing only averages may overlook essential details). Numbered 1 through 24 dice A set of four dice using all of the numbers 1 through 24 can be made to be intransitive. With adjacent pairs, one