# A game of dice

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

Note by Eddie The Head
3 years, 10 months ago

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

- 3 years, 10 months ago

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

- 3 years, 10 months ago

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

- 3 years, 10 months ago

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

- 3 years, 10 months ago

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

- 3 years, 10 months ago

I choose c...c<a

- 3 years, 10 months ago

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.

- 3 years, 10 months ago

Though withoutvan answer its a beautiful question

- 3 years, 10 months ago

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

- 3 years, 10 months ago