# About Lowering Target Numbers

### From Spiders

How advantageous is it to lower the target number on a roll, at least before the Central Limit Theorem kicks in?

Fix a target number *t*. Let *a*(*n*, *m*) denote the probability that a roll of *n* dice returns at least *m*;, where we put a(0, m) = 1 for m = 0 and 0 otherwise. Thus the cumulative distribution for the result from a roll of *n* dice is 1 - *a*(*n*, .). The generating function for *a* is given by

where α = (*t* − 1) / 10,β = 1 − *t* / 10, and γ = 1 / 10. With *p* valid charms such as World-Shaping Artistic Vision and Fateful Excellency that reduce the target number by 1, we have *t* = 7 - *p*. For fixed *n* and *p*, let *Q(d)* denote the quantile function for the outcome of the roll; that is, *Q(d)* is the minimum integer *m* such that *a*(*n*, *m*) <= 1 - *d*. The values of *Q(d)* for various values of *n*, *d*, and *p* = 7 - *t* are given below.

d = 0.9 | d = 0.75 | d = 0.5 | d = 0.25 | d = 0.1 | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

p | 0 | 1 | 2 | 3 | 0 | 1 | 2 | 3 | 0 | 1 | 2 | 3 | 0 | 1 | 2 | 3 | 0 | 1 | 2 | 3 |

n = 1
| 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

n = 2
| 2 | 2 | 3 | 3 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 |

n = 3
| 3 | 3 | 4 | 4 | 2 | 3 | 3 | 3 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 0 | 0 | 1 | 1 |

n = 4
| 4 | 4 | 4 | 5 | 3 | 3 | 4 | 4 | 2 | 2 | 3 | 3 | 1 | 1 | 2 | 2 | 0 | 1 | 1 | 2 |

n = 5
| 4 | 5 | 5 | 6 | 3 | 4 | 4 | 5 | 2 | 3 | 3 | 4 | 1 | 2 | 3 | 3 | 1 | 1 | 2 | 2 |

n = 6
| 5 | 6 | 6 | 7 | 4 | 5 | 5 | 6 | 3 | 4 | 4 | 5 | 2 | 2 | 3 | 4 | 1 | 2 | 2 | 3 |

n = 7
| 6 | 6 | 7 | 8 | 5 | 5 | 6 | 7 | 3 | 4 | 5 | 6 | 2 | 3 | 4 | 5 | 1 | 2 | 3 | 4 |

n = 8
| 6 | 7 | 8 | 9 | 5 | 6 | 7 | 8 | 4 | 5 | 6 | 6 | 3 | 3 | 4 | 5 | 2 | 2 | 3 | 4 |

n = 9
| 7 | 8 | 9 | 9 | 6 | 7 | 8 | 8 | 4 | 5 | 6 | 7 | 3 | 4 | 5 | 6 | 2 | 3 | 4 | 5 |

n = 10
| 8 | 9 | 10 | 10 | 6 | 7 | 8 | 9 | 5 | 6 | 7 | 8 | 3 | 5 | 6 | 7 | 2 | 3 | 4 | 6 |

n = 11
| 8 | 9 | 10 | 11 | 7 | 8 | 9 | 10 | 5 | 7 | 8 | 9 | 4 | 5 | 6 | 7 | 3 | 4 | 5 | 6 |

n = 12
| 9 | 10 | 11 | 12 | 8 | 9 | 10 | 11 | 6 | 7 | 8 | 10 | 4 | 6 | 7 | 8 | 3 | 4 | 6 | 7 |

n = 13
| 10 | 11 | 12 | 13 | 8 | 9 | 11 | 12 | 6 | 8 | 9 | 10 | 5 | 6 | 8 | 9 | 3 | 5 | 6 | 8 |

n = 14
| 10 | 12 | 13 | 14 | 9 | 10 | 11 | 13 | 7 | 8 | 10 | 11 | 5 | 7 | 8 | 10 | 4 | 5 | 7 | 8 |

n = 15
| 11 | 12 | 14 | 15 | 9 | 11 | 12 | 14 | 7 | 9 | 10 | 12 | 6 | 7 | 9 | 10 | 4 | 6 | 7 | 9 |

n = 16
| 12 | 13 | 14 | 16 | 10 | 11 | 13 | 14 | 8 | 10 | 11 | 13 | 6 | 8 | 9 | 11 | 5 | 6 | 8 | 10 |

n = 17
| 12 | 14 | 15 | 17 | 10 | 12 | 14 | 15 | 8 | 10 | 12 | 14 | 7 | 8 | 10 | 12 | 5 | 7 | 9 | 10 |

n = 18
| 13 | 14 | 16 | 18 | 11 | 13 | 14 | 16 | 9 | 11 | 13 | 14 | 7 | 9 | 11 | 13 | 5 | 7 | 9 | 11 |

n = 19
| 13 | 15 | 17 | 19 | 11 | 13 | 15 | 17 | 9 | 11 | 13 | 15 | 7 | 9 | 11 | 13 | 6 | 8 | 10 | 12 |

n = 20
| 14 | 16 | 18 | 19 | 12 | 14 | 16 | 18 | 10 | 12 | 14 | 16 | 8 | 10 | 12 | 14 | 6 | 8 | 10 | 13 |