Let’s talk about the elusive concept of “intransitivity” when it comes to dice rolling.

Before anything else: what is transitive? In simple terms, if A > B and B > C, then we can say that A > C (because if something is greater than another thing, and that other thing is also greater than a third thing, the original thing must be greater than that third thing too).

But wait what happens when this rule doesn’t hold true? That’s where intransitivity comes in. In some cases, we can have A > B AND B > C, but NOT necessarily A > C! This is called an “intransitive” relationship.

Now dice rolling. You might think that if you roll a six-sided die and get a five, then the next time you roll it, there’s a good chance you won’t get another five (because the probability of getting any specific number is equal). But what happens when we introduce intransitivity into the mix?

Imagine that instead of rolling one six-sided die, we have three dice let’s call them Dice A, B, and C. Each die has a different set of numbers on it:

Dice A: 1, 2, 3, 4, 5, 6 (standard)
Dice B: 1, 2, 3, 4, 5, 7 (with a “7” instead of a “6”)
Dice C: 1, 2, 3, 4, 6, 8 (with an “8” instead of a “5”)

Now let’s say we roll Dice A and get a five. Great! But what happens if we then roll Dice B? Well, since there is no “6” on this die, the probability of getting any specific number is still equal (1/6 for each). However…what if we then roll Dice C? Suddenly, our chances of rolling an eight are much higher than they were before!

This might seem like a silly example, but it actually has some interesting implications. For one thing, it shows that intransitivity can have real-world consequences for instance, when it comes to voting systems or economic markets. And for another thing…well, let’s just say that if you ever find yourself playing a game with these dice, be prepared for some unexpected twists and turns!

It might not sound like much at first glance, but trust us once you start thinking about it, the possibilities are endless (and sometimes downright mind-bending).