## Non-Transitive Dice

A couple of weeks ago, there was a general Spring cleaning at my workplace. In the pile of rubble waiting to be moved to the garbage dumpster, I saw a small container with 4 dice. I was intrigued.

It turns out the four dice were special dice called non-transitive dice.

The first die is red and has four sides with 2s and two sides with 6s. The second die is green and has three sides with 5s and three sides with 1s. The third die is purple and has four sides with 4s and two sides with 0s. The fourth die is yellow and has all six sides with 3s.

So, if you roll the red die and then the green die, there are four possible outcomes:

6-5 with probability = (2/6)(3/6) = 6/36 (red wins)
6-1 with probability = (2/6)(3/6) = 6/36 (red wins)
2-5 with probability = (4/6)(3/6) = 12/36 (green wins)
2-1 with probability = (4/6)(3/6) = 12/36 (red wins)

So red will beat green with probability 6/36 + 6/36 + 12/36 = 24/36 = 2/3. In other words, red is much better than green.

If you roll the green die and then the purple die there are four possible outcomes:

5-4 with probability = (3/6)(4/6) = 12/36 (green wins)
5-0 with probability = (3/6)(2/6) = 6/36 (green wins)
1-4 with probability = (3/6)(4/6) = 12/36 (purple wins)
1-0 with probability = (3/6)(2/6) = 6/63 (green wins)

So green beats purple with probability 24/36 = 2/3. Green is much better than purple.

If you roll the purple die and then the yellow die the two possible outcomes are:

4-3 with probability (4/6)(6/6) = 24/36 (purple wins)
0-3 with probability (2/6)(6/6) = 12/36 (yellow wins)

So purple is much better than yellow.

At this point, red beats green. Green beats purple. And purple beats yellow. You’d logically conclude that the red die must be much, much better than the yellow die. But if you roll the yellow die and then the red die:

3-2 with probability (6/6)(4/6) = 24/36 (yellow wins)
3-6 with probability (6/6)(2/6) = 12/36 (red wins)

In other words, yellow is much better than red!

This leads to a nice bar bet game. You let your friend pick any of the four non-transitive dice. Then no matter what die your friend picks, you can always select a better die. If you play several times, you’ll almost certainly win more times than you lose. Very cool practical application of math.