Non-transitive dice
Four unusual dice. You pick one, then I pick one of the others, and we roll: the higher number wins. Whatever you choose, I can choose a die that beats yours two times in three.
From the slides: Additional material.
Pick your die: click one of the four above.
Games
0
Your win rate
—
exact 1/3 = 0.3333
My win rate
—
exact 2/3 = 0.6667
Every pair, head to head
The chance that the row die beats the column die. There are no ties: no two dice share a face value.
What to notice
- A beats B, B beats C, C beats D, and D beats A — each with probability 2/3. “Beats” is not transitive.
- Check D against A by splitting into cases: D shows 5 (probability 1/2) and always wins; D shows 1 (probability 1/2) and wins only if A shows 0 (probability 1/3). Total 1/2 + 1/6 = 2/3. That case split is the law of total probability.
- The lesson for games: always let your opponent choose first.