Non-Transitive Dice
Main Concept
Imagine you are playing a game with a friend in which each of you chooses one die from the following set and roll them against each other to see who gets the higher face value.
Die
Faces
Expected Value
A
3
6
56⋅3 + 16⋅6 = 216=3.5
B
2
5
36⋅2 + 36⋅5=216=3.5
C
1
4
16⋅1 + 56⋅4 = 216=3.5
N (Normal)
16⋅1 + 16⋅2+16⋅3+16⋅4+16⋅5+16⋅6=216 = 3.5
If you let your friend choose his or her die first, is there a particular die you could choose from the remaining three to give yourself a better chance of winning? Which die would you choose if you were able to pick first?
The Solution
If your friend happens to choose one of the colored dice, A, B, or C, then yes, you can choose a certain die to give yourself the upper hand. If your friend happens to choose the normal die, N, then you will be playing with equal chances of winning, no matter which die you choose. The reason why competing with just the colored dice does not provide equal chances of rolling the higher number is because they are non-transitive dice —a special set of dice for which the property of "rolling the higher number more than half the time" is not transitive:
When rolling this set of dice repeatedly, you will see that A beats B most of the time, B beats C, and surprisingly, C beats A.
So, the fact that A beats B more than 50% of the time and B beats C more than 50% of the time DOES NOT ensure that A beats C more than 50% of the time!
A Look at the Probabilities
By observing all possible combinations found by rolling two of the dice, we can determine how often a certain die will roll a higher number than the competing die. In the case that we roll two of the non-transitive dice against one another, we will see that the probability of one of them winning is greater than 50%, meaning it should win more often. But, in the case that we roll a non-transitive die against a normal die, we will see that the probability of each of them winning is equal, meaning that they are equally matched and so should win the same amount of times.
A vs. B
Probability of A beating B = 2136≈ 58.3 %
Probability of B beating A = 1536 ≈ 41.7 %
Therefore, A is more likely to roll the higher number.
B vs. C
Probability of B beating C = 2136≈ 58.3 %
Probability of C beating B = 1536 ≈ 41.7 %
Therefore, B is more likely to roll the higher number.
A vs. C
Probability of A beating C = 1136≈ 30.6 %
Probability of C beating A = 2536 ≈ 69.4 %
Therefore, C is more likely to roll the higher number.
A vs. N
/
N
Probability of A beating N = 1536≈ 41.7 %
Probability of N beating A = 1536 ≈ 41.7 %
Probability of a tie = 636≈16.6 %
Therefore, A and N are equally likely to roll the higher number.
B vs. N
Probability of B beating N = 1536≈ 41.7 %
Probability of N beating B = 1536 ≈ 41.7 %
Therefore, B and N are equally likely to roll the higher number.
C vs. N
Probability of C beating N = 1536≈ 41.7 %
Probability of N beating C = 1536 ≈ 41.7 %
Therefore, C and N are equally likely to roll the higher number.
Use the check boxes below to choose two dice to roll, then press the "Roll!" button to roll them and see which one shows the higher face value. The Statistics plot will keep track of the number of "wins" for each competing die to allow you to observe whether or not one die is more likely to beat the other.
Alternatively, use the slider to choose the number of rounds to simulate and press the "Play" button to have the computer roll for you.
Die A
Die C
Die B
Die N
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