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A set of dice is nontransitive if it contains three dice, ''A'', ''B'', and ''C'', with the property that ''A'' rolls higher than ''B'' more than half the time, and ''B'' rolls higher than ''C'' more than half the time, but it is not true that ''A'' rolls higher than ''C'' more than half the time. In other words, a set of dice is nontransitive if the binary relation – rolls a higher number than more than half the time – on its elements is not transitive. It is possible to find sets of dice with the even stronger property that, for each die in the set, there is another die that rolls a higher number than it more than half the time. Using such a set of dice, one can invent games which are biased in ways that people unused to nontransitive dice might not expect (see Example). ==Example== Consider the following set of dice. * Die ''A'' has sides 2, 2, 4, 4, 9, 9. * Die ''B'' has sides 1, 1, 6, 6, 8, 8. * Die ''C'' has sides 3, 3, 5, 5, 7, 7. The probability that ''A'' rolls a higher number than ''B'', the probability that ''B'' rolls higher than ''C'', and the probability that ''C'' rolls higher than ''A'' are all 5/9, so this set of dice is nontransitive. In fact, it has the even stronger property that, for each die in the set, there is another die that rolls a higher number than it more than half the time. Now, consider the following game, which is played with a set of dice. # The first player chooses a die from the set. # The second player chooses one of the remaining dice. # Both players roll their dice; the player who rolls the higher number wins. If this game is played with a transitive set of dice, it is either fair or biased in favor of the first player, because the first player can always find a die that will not be beaten by any other die more than half the time. If it is played with the set of dice described above, however, the game is 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 5/9. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Nontransitive dice」の詳細全文を読む スポンサード リンク
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