A fun paradox arising from special dice. I'll decribe it in terms of a game.

You describe to your friend that you each pick a die, and roll them simultaneously. Whoever comes out with the highest number wins. You present the following dice (shown here unrolled):

    0        3        2        5
  4 0 4    3 3 3    2 2 2    1 1 1
    4        3        6        5
    4        3        6        5
--------------------------------------
    A        B        C        D

You graciously give him the "advantage" of choosing first. He chooses, then you choose from the remaining three. You play 100 games, and you have won about 66 games.

The secret is that you always pick the die to the left of the one he picked (in the diagram). If he picked A, you pick D. Why this is so counterintuitive is that people expect these kinds of things to obey transitivity, which they don't.

The expectation for each die is 2.66, 3, 3.33, and 3, respectively. This would make it seem that C is the optimal choice. Nonetheless, if you write out the tables of the 36 combinations of each pair, you will find that 24 of A's are greater than B's, 24 of B's greater than C's, 24 of C's to D's, and 24 of D's to A's! As such, if you go second, you can always pick a winning die.

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