Probably Not, 2nd Edition

13Two Paradoxes

Introduction

A paradox, as defined in dictionaries, is an “apparently false statement that proves to be true.” This chapter will present two such paradoxes: Parrondo's paradox and Simpson's paradox.

Parrondo's Paradox

In Chapter 8, we showed that an important factor in gambling games is the expected value of return. If the expected value isn't in your favor, you don't have to look at other factors; in the long run you should expect to lose money.

Juan Parrondo (a physicist), in 1966 described some games of chance in which properly combining two losing games produces a winning game. The first of these games is a simple board game where a player moves back and forth in one dimension, the direction of the move determined by the throw of a pair of dice. Figure 13.1 shows the playing board for this game and Table 13.1 shows two sets of rules for playing the game.

Figure 13.1 Playing board for Parrondo's paradox board game.

Table 13.1 Rule sets for Parrondo's Paradox board game.

Rule set 1		Rule set 2
White	Black	White	Black
Playing rules
Forward	7, 11	11	11	7, 11
Back	2, 3, 12	2, 4, 12	2, 4, 12	2, 3, 12
Probabilities of plays (36ths)
Forward	8	2	2	8
Back	4	5	5	4

We start using Rule Set 1, shown in the upper‐left section of Table 13.1. The player starts on the center (black) square and rolls the dice. The section of Rule Set 1 labeled ...