# About Lowering Target Numbers

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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

$f(x, y) = \sum_{n, m \geq 0} a_{n, m} x^n y^m = \frac{1 - \alpha x + \gamma xy}{(x - 1)(\alpha x + \beta x y + \gamma xy^2 - 1)} \in Q[[x, y]]$

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 22221111011100000000
n = 2 22332222111201110001
n = 3 33442333122211120011
n = 4 44453344223311220112
n = 5 45563445233412331122
n = 6 56674556344522341223
n = 7 66785567345623451234
n = 8 67895678456633452234
n = 9 78996788456734562345
n = 10 8910106789567835672346
n = 11 89101178910578945673456
n = 12 91011128910116781046783467
n = 13 101112138911126891056893568
n = 14 101213149101113781011578104578
n = 15 111214159111214791012679104679
n = 16 121314161011131481011136891156810
n = 17 1214151710121415810121478101257910
n = 18 1314161811131416911131479111357911
n = 19 13151719111315179111315791113681012
n = 20 1416181912141618101214168101214681013