# Relative Likelihood of n Sixes on 6n Dice

## Problem

Which is more likely:

- to throw at least $1$ six with $6$ dice
- to throw at least $2$ sixes with $12$ dice
- to throw at least $3$ six with $18$ dice?

## Solution

The chance of getting $1$ six and $5$ other outcomes in a particular order is $\paren {\dfrac 1 6} \paren {\dfrac 5 6}^5$.

We need to multiply by the number of orders for $1$ six and $5$ non-sixes.

Therefore the probability of exactly $1$ six is:

- $\dbinom 6 1 \paren {\dfrac 1 6} \paren {\dfrac 5 6}^5$

Similarly, the probability of exactly $x$ sixes when $6$ dice are thrown is:

- $\dbinom 6 x \paren {\dfrac 1 6}^x \paren {\dfrac 5 6}^{6 - x}$

for $x = 0, 1, 2, 3, 4, 5, 6$.

The probability of $x$ sixes for $n$ dice is:

- $\dbinom n x \paren {\dfrac 1 6}^x \paren {\dfrac 5 6}^{n - x}$

for $x = 0, 1, 2, \ldots, n$.

according to the binomial distribution.

The probability of $1$ of more sixes with $6$ dice is the complement of the probability of $0$ sixes:

- $1 - \dbinom 6 0 \paren {\dfrac 1 6}^0 \paren {\dfrac 5 6}^6 \approx 0.665$

When $6 n$ dice are rolled, the probability of $n$ or more sixes is:

- $\ds \sum_{x \mathop = n}^{6 n} \dbinom {6 n} x \paren {\dfrac 1 6}^x \paren {\dfrac 5 6}^{6 n - x}$

which equals:

- $1 - \ds \sum_{n \mathop = 0}^{n - 1} \dbinom {6 n} x \paren {\dfrac 1 6}^x \paren {\dfrac 5 6}^{6 n - x}$

Calculating the probabilities, we get:

$6 n$ $n$ $\map P {\text {$n$ or more sixes} }$ $6$ $1$ $0.665$ $12$ $2$ $0.619$ $18$ $3$ $0.597$ $24$ $2$ $0.584$ $30$ $5$ $0.576$ $96$ $16$ $0.542$ $600$ $100$ $0.517$ $900$ $150$ $0.514$

It is clear that the gambler does better by betting on $1$ six on $6$ dice than $2$ on $12$ or $3$ on $18$.

$\blacksquare$

## Historical Note

According to David Wells in his *Curious and Interesting Puzzles*, this question was asked by the famous diarist Samuel Pepys of Isaac Newton in $1693$.

## Sources

- 1987: Frederick Mosteller:
*Fifty Challenging Problems in Probability with Solutions*(2nd ed.) - 1992: David Wells:
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