Roll Twice Take Highest

The generalizable way to get the mean value of a rerolled N-sided die (taking the highest result):

(1)\begin{align} \sum_{n=1}^{N} \frac{2n^{2}-n}{N^{2}} = \frac{N(N + 1)(4N - 1)}{6N^{2}} \end{align}

… and the mean value 'R' of a re-rolled 'n' on a N-sided die:

(2)\begin{align} R(n) = \frac{n^{2} - n + N^{2} + N}{2N} = \frac{n^{2} - n}{2N} + \frac{N + 1}{2} \end{align}

Examples:

dX | mean reroll | mean 2x | reroll >2x for |
---|---|---|---|

d3 | 2.444 | 4 | 1 |

d4 | 3.125 | 5 | 1 |

d6 | 4.472 | 7 | 1-2 |

d8 | 5.8125 | 9 | 1-2 |

d10 | 7.15 | 11 | 1-2 |

d12 | 8.486 | 13 | 1-3 |

## What About the Lowest of Two Rolls?

The generalizable way to get the mean value of a rerolled N-sided die (taking the lowest result):

(3)\begin{align} \sum_{n=1}^{N} \frac{Nn}{N^{2}} - \frac{n(n - 1)}{2N^{2}} = \frac{N(N + 1)(2N + 1)}{6N^{2}} \end{align}

page revision: 2, last edited: 13 Jun 2012 14:38