You are given 3 dice as shown in the figure above where the numbers on the opposite faces of the each of the dice are equal.(For example,the number opposite to 6 in the first dice is also 6.)

Now the game begins like this I choose 1 dice and you choose another one from the remaining and after rolling whichever comes up with larger value wins the game.By choosing which dice are you more likely to win(that is with greater probability) in the following cases:

A.I choose dice A.

B.I choose dice B

C.I choose dice C

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## Comments

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TopNewestA beats B, B beats C, and C beats A, all with 5/9 probability. Nontransistive dice, i.e, A > B > C > A

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Haha....indeed....but rolling twice the case becomes entirely opposite.....but does a pattern hold if we generalize it to k rolls?

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I believe non-transitive dice were discovered, if not just made popular, by Dr. James Grime (seen on numberphile)?

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Yeah...I remember seeing that on numberphile too....

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A. I will choose dice C B. I will choose dice A C. I will choose dice B

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I choose c...c<a

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A is actually no better than B, and B is no better than C. If u calculate the expected values, they are just the same. This means it will be a fair bet between any chosen pair of dice if u include the rewards for winning a bet as the difference in numbers rolled on the dice. When we choose A to bet with B, for instance, it is possible to get more rewards on B (example: u roll a 9) even when there is a lower probability (4/9) to win.

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Though withoutvan answer its a beautiful question

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If you want the answer you may choose to read on non-transitive dice

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