## December 22, 2008

### The Statistics of 4E Skill Challenges

A friend of mine recently said the following on a private forum I'm a member of ...
" ...I may be thinking about this wrong, but let's go with this - DC 30 with 6 rolls required before 2 failures. Assuming a +15, you have a 50% chance of a success per roll and a 50% chance of failure. So what's the chance you'll fail before you make it? Let's just go with the stats of what it would be if you got to roll all 6 no matter what, which is approx. 25%. The SAME thing could be accomplished by having someone with a +15 roll against a DC35..." Anonymous Friend.
I just couldn't help myself from chiming in when I read this. Skill challenges in 4th Edition Dungeons & Dragons are not easily reduced to equivalent single-roll "skill checks". They are more aptly represented by a collection of negative binomial distributions. This is likely technobabble to many of you, but suffice it to say that skill challenges represent a slight modification of well known type of probability distribution. Negative binomial distributions are usually those that describe a situation where you want to measure the probability of X number of successes before you reach Y number of failures given a probability of success P. The "slight modification" comes in because you are also considering that you might get all your successes before having any, or only some, of your failures.

I used Google Spreadsheet's bionomial normal distribution function to make these calculations by summing three distributions together that each had a different threshold for failure (0, 1, or 2 failures) like this:
=NEGBINOMDIST(X,N,P)+NEGBINOMDIST(Y,N,P)+NEGBINOMDIST(Z,N,P)
where N is the number of successes needed, X, Y, or Z are the number of failures observed (0, 1 or 2), and P is the probability of success for the skill checks.

So, in the table below I have outlined the probability of success for a skill challenge that requires N number of successes before 3 failures. Across the top of the table you can see the actual d20 die roll needed for the success, and along the left you can see the total number of successes needed for the entire skill challenge to be a success. I'm also assuming that three-failures always represents failure of the skill challenge as a whole. Of course, this is still a simplification because I'm also assuming that the all the rolls would have the same DC - which of course is rarely the case. The table is also incomplete, but it gives you the general idea: skill challenges quickly become very hard as the complexity increases.

UPDATED -- 1/9/09 -- UPDATED

Use the above table to give yourself a rule of thumb when designing skill challenges. What's important to realize is that Complexity 5 skill challenges (i.e. 12 successes before 3 failures) are extremely hard even when needing very low rolls. I have no doubt that this is the reason why the DCs for skill challenges were made to be so low after the 4th Edition Errata was released for the Dungeon Masters Guide. I also believe this is a point many DMs miss.

Hopefully this post has shed some light on why the difficultly of skill challenges can be tricky to guage, and possibly helped a few folks in the meantime.

1. Wait, so just to make sure I am reading your table properly -

Each cell effectively represents, given the average roll needed to meet or beat the average DC of the skill checks involved (columns), the likelihood of beating a skill challenge requiring a given number of successes (rows)?

thanks for doing this. yay stats!

2. yeah, basically.

So.. for example .. if you had a Complexity 2 skill challenge with moderate difficulty (you need to roll a 10 or higher).. thats 6 successes before 3 failures with a 55% chance of success for each roll..

but overall.. there's only a 22% chance of success for the whole Skill Challenge.

3. Yes... I think one of the mistakes GMs make with skill challenges is assuming that the PCs are meant to win them. This isn't necessarily the case. Failing a skill challenge isn't supposed to roadblock the game, it's supposed to introduce complications and obstacles for the PCs to deal with.

Another thing I think people miss is that bonuses are meant to be reasonably easy to come by, either mechanically (use of skill X gives a +Y bonus to the next use of skill Z) or through "circumstance bonuses" for good roleplaying and clever ideas. The bonuses make it much easier to succeed at the challenge, and therefore reward said good roleplaying and clever ideas.

I suspect I'll be posting about skill challenges soon; I have some other thoughts on them.

4. Hey Scott, thanks for stopping by! I'm looking forward to your take on the skill challenges - should be good.

5. I just realized that there is a a flaw in my statistical analysis of this - the special cases where the number of successes needed are less than the number of failures (1 or 2) successes should use a different formula that does not include the 2nd or 3rd rolls. Once I get around to fixing this this, I'll update the table. So, for now, ignore the first two rows of the table, the math is wrong; but the table is correct for S.C. that need 3 or more successes. Ahhh silly me...

6. Is this worse than the original system of failures equal to half the numbers of successes?

7. @Anonymous : Yes it is, however they also lowered all the DC's needed to count as successes (Easy, Moderate, Hard) - this is why some people were confused when they lowered the DC's but the S.C.s still seemed to be just as hard. You fail faster - but probably with about the same rate of failure as compared to the old system with the higher DCs. I haven't honestly looked analytically at this; but this is what my gut expects to the case.

8. I came up with my own DCs, which I think are good for individual checks. Closer to the new DCs plus 5 than the old DCs without the additional 5. I hadn't considered the new DCs went along with the 3 failures, which I understood fixed higher complexities being easier and I found was a lot easier to play out - players didn't need to know the complexity going in to determine how many failures they had left, and thus how bold or cautious they needed to be.

I took the highest skill modifier for each skill of my party, the average of was +12.5 at 7th Level. My Moderate DC for that level was 19, so let's say they needed a 6 die roll. Looking at your table, that's cool for Complexity 1, 2 and even 3. It comes off the rails after that. The new DCs Moderate for that level is 14, so let's say they needed a 2 die roll. Looking at your table, that's not a challenge.

Not sure which direction to go in now.

9. I've fixed the problem (dropping the first two rows, becuase the minimum skill challenge is three successes if you allow for 3 failures). I also found out that there was a bug in the calculations... so ...

the table is all new and updated. Should be correct now.