Number Crunching: Getting Some Boring Stuff Out Of The Way

How much XP does it take to get to Level n?

XP(n) = 50 n2 - 100 n + 50 or = 50 (n-1)2

Before we can begin addressing the key questions, we need to answer some more basic questions first. The "Statistical Analysis..." thread already discussed the theory behind these, so we can be a bit more practical here.This is pretty straightforward. The Experience and Fame page on the wiki gives a table of XP values, and it was easy to put them into Excel and generate a curve that solves this precisely:

For example, newly-created Level 1 characters have XP(1)=0 XP, while fresh-20 characters have at least XP(20)=18,050 XP.

Just How Good Is A Roll Of h At Level n?

Again, the “Statistical Analysis…” thread discusses this. Instead of dealing with really friggin’ big numbers here, I just generated a spreadsheet that calculated probabilities of having exactly a roll of h at Level n, then used that to generate a page of cumulative percentile ranks of each roll at each level:

We can interpret this table as follows: at Level 9, a +5HP roll is in the top 29.1577% of all rolls at that point. At level 20, a -2HP roll is in the top 55.72% of all rolls (values greater than 50% mean it’s below average, which is what we’d expect for a negative roll).

At any level, a roll of ±0HP is in the top 50% of all rolls—average.

We generally only care about percentile ranks at Level 20. It’s enticing to think “Oh, I have a top 10% roll at Level 10, surely it’ll turn into a top 10% roll at Level 20!”, but probability doesn’t work that way. In fact, if you have a roll of h at any midway point, there’s a very good chance that your roll at Level 20 won’t be too far from that. That’s just how the random numbers work here.

So, given a desired final roll of h, we can define p as the percentile rank of that roll at Level 20. This sheet serves as a lookup table for those values.

Okay...so if My Character Is Level n, What Is The Probability That His Roll Will Increase By At Least ΔHP At Level 20?

I am so glad you asked.

At Level 19, we only have one level-up remaining (so your character’s roll can’t change by more than 5HP in either direction). At Level 9, we have eleven level-ups remaining. At any Level n, we have 20-n level-ups remaining. So, we can flip the above tables around, and calculate the probability of gaining at least h HP between now and Level 20:

We can interpret this table as follows: if my character is Level 9, there is a 33.5938% chance that his roll will improve by at least +5HP by the time he hits Level 20. If my character is Level 19, there is a 9.0909% chance that his HP will improve by +5HP when he hits Level 20 (a 1/11 chance). At Level 11, there is a 71.6051% chance that he won’t lose more than -5HP off his roll (again, an over-50% chance means below average—in this case, your roll got worse).

At Level 20, there is a 100% chance that his HP won’t improve any more. (Duh.)

Given where your character is at Level n, we normally look at this table in terms of “desired change in HP”. If we want +15HP and we’re currently at +3HP, that’s a change of 15-3=12HP. If we want +5 HP and we’re currently at -2HP, that’s a change of 5-(-2)=7HP. Let’s define this “desired change in HP” as ΔHP.

From this, we can define P(ΔHP,n) as the probability that our character’s HP roll will increase by at least ΔHP between Level n and Level 20. This sheet serves as a lookup table for those values.