Cumulative distribution function¶

How do we start analysing the probabilities for high runes? We need a statistical model to explain this random phenomenon. The distribution that should apply quite well for our purposes is the geometric distribution. In this instance, we specifically want to analyse how many LK runs (attempts or trials) we should expect and what kind of variation in the number of runs there will be - statistically. We don't just want to a single number representing how many runs are needed on average, but also to define some kind of a range, or a lower and upper limit of runs that can be reasonably expected.

I'm not going to explain how the mathematics work, but I will say this:

The geometric distribution gives the probability that the first occurrence of success requires $k$ independent trials, each with success probability $p$.

This applies to Diablo pretty well, since each trial is indeed independent of each other. Whether or not you found a SoJ on your previous Mephisto run does not affect the probability on your next run. Also, generally in Diablo we are concerned about the number of runs $k$ (trials) as a metric for how much effort or time is needed to get an item.

The cumulative distribution function (CDF) of a random variable $X$ tells us the probability that $X$ is less than or equal to a given value. For example, the CDF of the geometric distribution will tell us:

What is the probability to get a successful drop in less than or equal to 100 LK runs?

Isn't that nice? Yes, indeed. (Read the wikipedia pages if you want to know more.)