A Treatise on Sucker's Bets

TL;DR

This paper explores non-transitive probability scenarios using custom dice and card decks, demonstrating how to identify and analyze 'sucker's bets' with computational methods, revealing their properties and likelihoods.

Contribution

It introduces a systematic approach to find and analyze non-transitive probability examples using computer algebra systems, extending the concept beyond dice to decks of cards.

Findings

Identified all non-transitive examples with specified decks and sizes.

Developed an automated method to compute probabilities of sucker's bets.

Provided statistical insights into the likelihood of non-transitive relations.

Abstract

In 1970, Statistics giant, Bradley Efron, amazed the world by coming up with a set of four dice, let's call them A,B,C,D, whose faces are marked with [0,0,4,4,4,4], [3,3,3,3,3,3],[2,2,2,2,6,6],[1,1,1,5,5,5] respectively, where die A beats die B, die B beats die C, die C beats die D, but, surprise surprise, die D beats die A! This was an amazing demonstration that "being more likely to win" is not a transitive relation. But that was only one example, and of course, instead of dice, we can use decks of cards, where they are called (by Martin Gardner, who popularized this way back in 1970) , "sucker's bets". Can you find all such examples, with a specified number of decks, and sizes? If you have a computer algebra system (in our case Maple), you sure can! Not only that, we can figure out how likely such sucker bets are, and derive, fully automatically, statistical information!