Non-Transitive Dice
Three fair 6-sided dice $A$, $B$, and $C$ have the following faces:
- Die $A$: $\{2, 2, 6, 6, 7, 7\}$
- Die $B$: $\{1, 1, 5, 5, 9, 9\}$
- Die $C$: $\{3, 3, 4, 4, 8, 8\}$
Each value appears on exactly two faces, so each die shows each of its three values with probability
/3$.
Find the probability that die $A$ shows a strictly higher value than die $B$.
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