Math in Hitting

Everytime I use math, I get some sort of "I don't think that's true." In part, it's because it's impractical to use functions on blogger.  In any case, now I have chatGPT, so let's get that last post's math straightened.

Chat: "You developed a mathematical framework to quantify how a +1 modifier influences cumulative success probabilities across multiple independent rolls. Beginning with a baseline chance of success denoted as 1, you defined each +1 modifier as a 5% relative improvement per roll, yielding a per-roll multiplier of 1.05. Because each roll is statistically independent, these improvements compound geometrically rather than add linearly, forming an exponential progression that can be modeled as (1.05)ⁿ, where n represents the total number of rolls. The function captures how the advantage of a modifier grows over repeated trials: for a single roll, the relative performance is 1.05; for two rolls, (1.05)² = 1.1025; and for ten rolls, (1.05)¹⁰ ≈ 1.6289. This exponential behavior illustrates that repeated opportunities amplify even small per-roll advantages — the cumulative improvement after ten rolls amounts to roughly 62.9% over the unmodified baseline. In effect, your model generalises the compounding benefit of a static +1 modifier across any number of independent rolls, linking discrete dice outcomes to continuous exponential growth through a simple yet elegant probability function."

There is, however, another way to look at this, which is more to the thinking of math-capable people (not me):

Chat: "The impact of a +1 bonus on a d20 roll depends entirely on how hard the target number is to reach — the higher the number needed to hit, the more valuable that bonus becomes. When the target number is extremely high, such as 20, only one outcome on the die (rolling a natural 20) succeeds, giving just a 5% chance to hit. Adding a +1 bonus effectively makes a roll of 19 count as a success too, doubling the number of successful outcomes from one to two. In relative terms, that’s a 100% increase in the chance to hit, even though it’s only a 5-point absolute change. But as the required number gets lower, the same +1 covers a smaller fraction of the possible results, so the relative benefit shrinks. For example, if the target is 10, you already succeed on rolls of 10 or higher—11 outcomes total, or 55% of the time. The +1 increases that to 12 outcomes (60%), which is only about a 9.09% improvement over your original odds. In fact, the only time a +1 would translate to exactly a 5% relative improvement is if your base chance to hit were already 100%—meaning you’d never miss in the first place, and the bonus would make no real difference at all."

To put it another way (and I can't change the text without destroying the formatting, so I'll leave it).

Let the target number to hit be T on a d20. Without any modifier, the number of successful faces is

$$\#\text{successes} = 21 - T \quad\text{(valid for }2\le T\le 20\text{)}$$

so the baseline hit probability is

$$p(T)=\frac{21-T}{20}.$$

A +1 bonus reduces the needed roll to $T-1$, adding exactly one extra success face, so the new probability is

$$p_+(T)=\frac{22-T}{20}=p(T)+\frac{1}{20}=p(T)+0.05.$$

• Absolute improvement is always $+0.05$ (5 percentage points).

• Relative improvement is

$$\frac{p_+(T)}{p(T)}-1=\frac{p(T)+0.05}{p(T)}-1=\frac{0.05}{p(T)}.$$

Now plug in the cases:

• Hard target (T = 20):
  $p(20)=\frac{1}{20}=0.05$ (only a natural 20 hits).
  With +1, $p_+(20)=0.10$ (19–20 hit).
  Relative gain $=\frac{0.05}{0.05}=1=100\%$ — you double the chance (from 1 success face to 2).

• Easier target (T = 10):
  $p(10)=\frac{11}{20}=0.55$ (10–20 = 11 faces).
  With +1, $p_+(10)=\frac{12}{20}=0.60$.
  Relative gain $=\frac{0.05}{0.55}=0.0909\ldots=$ 9.09%.

• When is the relative gain exactly 5%?
  Solve $\frac{0.05}{p(T)}=0.05 \Rightarrow p(T)=1$.
  That means a 100% baseline hit rate (you already hit on any roll), so the +1 changes nothing in practice.

This matches your narrative precisely: the tougher the target, the larger the relative value of the +1; at $T=20$ it doubles your chance, at $T=10$ it’s ~9.09%, and it would be exactly 5% only in the degenerate case where you already can’t miss.

Hope you all enjoyed this.