Nontransitive dice

Intransitive dice is called a set of special dice, in which there is to each of the cube to another cube, he will lose in the long run against, that is, compared with which he often shows a smaller than a larger number. An example is shown at right three intransitive dice A, B and C: each with probability 5/9 A wins against B, B against C and C to A. The example of intransitive dice shows that the relation " is more likely to be greater " for random variables need not be transitive. A similar example of an intransitive relation is the game scissors, paper, stone, in which each symbol wins against one and lose to another.

The result of the game is counter-intuitive that an advantage must be transitive. This notion would be true if the result would be the sum of diced in a large number of game rounds numbers and not the number of rounds won. A similar mistake is the Condorcet paradox.

Efron's dice

Efron's dice are four intransitive dice, which were invented by the American statistician Bradley Efron.

The four cubes A, B, C and D have the following numbers on their eyes six pages:

• 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

For each of the cube, there is another one who defeated him with probability 2/3:

• P (A> B ) = P ( B> C) = P (C > D ) = P ( D> A) = 2/3.

The probabilities of the Rates of A with C, and B with D

• P (A > C) = 4/9 and P ( B> D) = 1/2.

Miwin'sche cube

The Miwin'schen cubes were invented in 1975 by the Austrian physicist Michael Winkelmann. They are labeled as follows:

Set 1

• III: 1, 2, 5, 6, 7, 9
• IV: 1, 3, 4, 5, 8, 9
• V: 2, 3, 4, 6, 7, 8

Set of 2

• IX: 1, 3, 5, 6, 7, 8
• X: 1, 2, 4, 6, 8, 9
• XI: 2, 3, 4, 5, 7, 9

Against each of the cube has one of the other two following opportunities: profit 17/36, loss of 16/36 and Drawn 3 /36. Winkelmann has also constructed intransitive dice in dodecahedral form.