Three dice with no best die

Question

Three special six-sided dice are on the table.

Die A shows $3$ on every face.
Die B shows $2$ on four faces and $6$ on two faces.
Die C shows $1$ on two faces and $5$ on four faces.

Two players each roll one die, and the higher number wins. Your friend gets to pick a die first, and then you choose one of the remaining two. Your friend picks die A.

Which die should you take, and what is your probability of winning?

Accepted Answer

Check who beats whom. - A versus B: A always shows 3. B shows 2 (probability 	frac46) losing to A, or 6 (probability 	frac26) beating A. So A beats B with probability 	frac46 = 	frac23. - B versus C: rolling both out, B beats C with probability 	frac{20}{36} = 	frac59. - C versus A: A always shows 3. C shows 5 (probability 	frac46) beating A, or 1 (probability 	frac26) losing. So C beats A with probability 	frac46 = 	frac23. The dice form a cycle: A beats B, B beats C, and C beats A. There is no strongest die. Your friend chose A, and C beats A. So take die C. It shows 5 two-thirds …

Three dice with no best die

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