The Dice Problem

Origin

The Problem of Dice dates back at least to the $17$th century and has since taken many forms. It is one of the problems that Blaise Pascal and Pierre de Fermat discussed which helped them form rules for basic probability theory.

Problem and Solutions

This is the variation considered here:

Is it better to try for one six in four tosses of a single die, or try for a pair of sixes in $24$ tosses of a pair of dice?

The Chevalier de Mere considered the problem and thought he found a contradiction in algebra. He noted that there are $6$ possible outcomes for one toss of a die, and $36$ for the toss of a pair of dice, and in each case, there is exactly one way to observe the desired outcome. He argued that since the ratio $4:6$ is equivalent to $24:36$, the probability in either case should be the same. However, he also calculated the probabilities as follows:

$$P( \text{at least one six in four tosses of a single die})$$ $$=1-P(\text{no $6$s in four tosses})=1-(5/6)^4 \approx 0.5177.$$ and $$P(\text{at least one pair of sixes in $24$ tosses of a pair of dice})$$ $$=1-P(\text{no pairs of sixes})=1-(35/36)^{24} \approx 0.4914.$$

Pascal noted that the probability calculations were correct, but de Mere's argument regarding the ratios was invalid because no part of the probability calculations shows a proportional relationship between the two (notice that the $6$ is raised to the fourth power in $(5/6)^4$ and the $36$ is raised to the $24$th in $(35/26)^{24}$). Such a relationship between does not imply proportionality.

Since the probability of at least one six in four tosses of a single die is about $0.5177$ and the probability of at least one pair of sixes in $24$ tosses of a pair of dice is about $0.4914$, it is slightly better to try for one six in four tosses of a single die.