Consider three nonstandard dice. Instead of the numbers 1 through 6, die A has two 3??s, two 5??s, and two 7??s; die B has two 2??s, two 4??s, and two 9??s; and die C has two 1??s, two 6??s, and two 8??s, as in Figure 2.8.?

Suppose dice A and B are rolled. Show that A is more likely to get the higher number. That is, P(A > B) > 0.50, where {A > B} denotes the event that A beats B.?

Now show that if B and C are rolled, B is more likely to get the higher number. And, remarkably, if C and A are rolled, C is more likely to get the higher number.?

Many relationships in life are transitive. For instance, if Amy is taller than Ben and Ben is taller than Charlie, then Amy is taller than Charlie. But these dice show that the relation ??more likely to roll a higher number?? is not transitive.?

The dice are the basis of a magic trick. You pick any die. Then I can always pick a die that is more likely to beat yours. If you pick A, I pick C. If you pick B, I pick A. And if you pick C, I pick B.

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8 3 7 3 2 49 1 6 6 9. 8 B 4. A,