Leith

1357482. Mon Aug 31, 2020 10:29 am 


A topic brimming with possibilities for confounding the intuition.
Lets design some dice. To start with we have some blank cubes onto which we can write some numbers. They don't have to be the traditional 1 to 6, nor do they need to be distinct, so we could have e.g. 1,1,2,2,3,3 or 0,4,8,10,15,20.
Say we have three dice, A, B and C.
We'll design them such that:
 Die A is likely to roll higher than B (i.e. A will roll higher than B more than half the time).
 Die B is likely to roll higher than C.
So, which of dice A and C is likely to roll higher?
If you're thinking that A beats B, and B beats C, therefore A must beat C, then await the klaxon.
In fact from the information above, we can't tell which of dice A and C will likely roll higher.
Further, it's possible to design sets of dice such that for any die in the set, there is another die that is likely to roll higher.
Here's an example set of four, known as Efron's dice, in which die A will roll higher than die B with odds of 2 to 1 and, with the same odds, B beats C, C beats D, and D beats A:
A: 4, 4, 4, 4, 0, 0
B: 3, 3, 3, 3, 3, 3
C: 6, 6, 2, 2, 2, 2
D: 5, 5, 5, 1, 1, 1
https://en.wikipedia.org/wiki/Nontransitive_dice 
