Context: There are a number of posts on this forum (not just this board) about growth rates and assertions about their related probabilities. They are typically in the form of
Equivalently stated, the assertion that is being made is that a single point of growth, over the course of 100 level-ups, is worth one stat point. While intuitive, this is not a true statement and is not a good way to think about growth related probabilities.
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Skimming over a few details, a Bernoulli random variable is a function which maps to either 0 or 1, according to some probability density function. In simpler terms, we we can say that P(X=1) = 0.25 if there is a 25% chance that random variable X maps to 1. (You can equivalently define X by stating that P(X=0) = 0.75.)
Suppose we are tossing a fair coin. We define random variable X to be the act of tossing the coin, and a heads is represented by a 0 and a tails is represented by a 1. Then, P(X=0) = P(X=1) = 0.5, and thinking about a single toss is straightforward. Now we have a way of formally asking, "when unit X levels up, how frequently do they a point of speed?".
Consider a game where we toss a coin 10 times and a success is defined as the coin landing on heads. The number of possible successes is [0, 10] inclusive. Let G be a random variable which describes the outcome of playing this game. We can define the probability of G as the summation of X over 10 trials, where P(X=1) = 0.50. Now we can formally ask, "when unit X levels up 10 times, how frequently will they gain exactly m points of speed?" by computing P(G=m). We can also ask more useful questions such as, "when unit X levels up L times, how frequently will they gain at least m points of speed?" by computing P(G_L>=m), where G_i is the random variable of the coin tossing game over i throws.
That was a mouthful, but we are almost there! A binomial distribution is defined as the sum of Bernoulli random variables -- in other words, the sum of evaluating a Bernoulli random variable over some number of trials. (Formally there are two requirements about those variables, namely that they be independent and identically distributed, but this is not important in the context of this discussion!)
Okay, so what does that mean for us?
A unit, U_1, has a speed growth of 0.5. Over 10 levels, the probability that they will gain at least 5 points of speed is roughly 62%. In notation, P(G_U1>=5) = 0.623, the probability that over 10 Bernoulli trials, there will be at least five successes.
A unit, U_2, has a speed growth of 0.4 (i.e., the same unit is a Paladin). The probability that they will gain at least 4 points of speed is roughly 61%. Additionally, P(G_U2>=5) = 0.36, whereas P(G_U1>=6) = 0.38.
The further you deviate from the center of the distribution, the more extreme these differences will become. The intuitive case is when you go from a growth of 0.1 to a growth of 0.0 over any number of level-ups. I encourage you to experiment with these numbers on your own, as probability can be extremely unintuitive.
https://www.wolframalpha.com/examples/mathematics/probability/bernoulli-trials/
Thanks for reading, this is a pet peeve of mine.
tl;dr: 1% of growth != 1 stat gain over 100 level-ups, and stat growths can be perfectly modeled using Bernoulli trials or a binomial distribution.
