biased in favor of the second player, because the second player can always find a die that will beat the first player's die with probability . The following tables show all possible outcomes for all three pairs of dice. Comment regarding the equivalency of intransitive dice Though the three intransitive dice A, B, C (first set of dice) A: 2, 2, 6, 6, 7, 7 B: 1, 1, 5, 5, 9, 9 C: 3, 3, 4, 4, 8, 8 P(A > B) = P(B > C) = P(C > A) = and the three intransitive dice A′, B′, C′ (second set of dice) A′: 2, 2, 4, 4, 9, 9 B′: 1, 1, 6, 6, 8, 8 C′: 3, 3, 5, 5, 7, 7 P(A′ > B′) = P(B′ > C′) = P(C′ > A′) = win against each other with equal probability they are not equivalent. While the first set of dice (A, B, C) has a 'highest' die, the second set of dice has a 'lowest' die. Rolling the three dice of a set and always using the highest score for evaluation will show a different winning pattern for the two sets of dice. With the first set of dice, die B will win with the highest probability () and dice A and C will each win with a probability of . With the second set of dice, die C′ will win with the lowest probability () and dice A′ and B′ will each win with a probability of . Variations Efron's dice Efron's dice are a set of four intransitive dice invented by Bradley Efron. The four dice A, B, C, D have the following numbers on their six faces: A: 4, 4, 4, 4, 0, 0 B: 3, 3, 3, 3, 3, 3 C: 6, 6, 2, 2, 2, 2 D: 5, 5, 5, 1, 1, 1 Probabilities Each die is beaten by the previous die in the list, with a probability of : B's value is constant; A beats it on rolls because four of its six faces are higher. Similarly, B beats C with a probability because only two of C's faces are higher. P(C>D) can be calculated by summing conditional probabilities for two events: C rolls 6 (probability ); wins regardless of D (probability 1) C rolls 2 (probability ); wins only if D rolls 1 (probability ) The total probability of win