Statistics Help

[subplotkudzu]

Can anyone give me a Plug-Into-Excel level formula for determining the overall percent chance of success of a test where the goal is to get 8 successes before 4 failures (or, put another way, to get 8 successes out of 11 rolls)?

 

For example: a 60% chance must occur 8 times in 11 rolls. What is the % chance this will happen? 

 

Obviously I’m trying to work out the overall percentages on the D&D 4E Skill Challenges. My hope is to learn what the designers intended as the final chance of success and use that as part of my framework for reverse engineering the idea into D&D 3E. It’s a good idea and worth stealing (even if I don’t care for the way it’s implemented in 4E).

 

Here’s the 4E implementation: the Skill Challenge is measured in its Level (compared to the PCs level to determine Difficulty Class for roll) and Complexity (the number of successes required & number of failures allowed to complete the test; each level of complexity counts as 1 at-level monster in an encounter (which is a pretty damn sweet mechanic)) Every PC is supposed to have rolls in an initiative sequence to contribute to the challenge.

 

My problems with it: First, I don’t like the Level being measured against PC Level  as it sways too far away from simulation for me, and the difficulty options are too limited (easy, medium, hard).  Second, the scaling advice makes it feel like the PCs aren’t ever improving even as their numbers change. Third, I have no adequate way to gauge overall chance of success in the “Make 12 successes before 6 failures” mechanic. Finally, I dislike the idea of forcing each character to make a roll in every round of the skill challenge – I understand the rationale of keeping everyone involved, but my players have no problem ceding spotlight time to the party experts.

 

The fourth problem is one that I can deal with simply by continuing to play as I do.  The statistics question at the top of this post should resolve the third problem. For the first and second I have been working out my alternate system using the 3E challenge ratings as a scaling point.

 

I’m basing this on the following piece of logic:

1) Skill Challenges in 4E have equivalences with Monsters – a complexity 1 challenge can take the place of one monster at a PC level challenge rating in an encounter.

2) A monster with a PC level challenge rating in 4E is meant to be a challenge to one PC.

3) A challenge rating of 1 in 3E means that a single monster is meant to be a challenge for a party of 4 PCs.

4) Therefore what would be a Challenge Rating 1 monster in 4E is a Challenge Rating 1/4 monster in 3E.

5) And thus Challenge rating 1/4 monster would be equal to a challenge rating 1/4 Skill Challenge in 3E.

So how do we define this?

 

3E sets the level of an Encounter Level based on the CR of an individual foe and the number of foes there are. Four CR 1/4 Goblins make an EL 1 encounter. It’s easy enough to see the 4E Level (how high you need to roll) as the 3E Challenge Rating and the 4E complexity (how many successes you need) as the number of monsters. Make sense?

 

Unfortunately combat equivalences get difficult, because there are so many more factors in combat: opponent attacks, what your AC is, what your HP are, what your attack bonus is, how much damage you do and all of those factors on the other side when they attack you.  Based on some combat modeling as well as personal experience I believe it takes about 5 rounds of combat to conclude an At Level encounter. For 4 PCs that’s 20 attack opportunities , of which they will score about 50% successes (some successes will have 2-4 times normal effect based on damage rolls, so not everyone has a 50% to hit rate). An at-level encounter is meant to drain the PCs of 20% of their resources, not kill them. So failure in an at-level encounter doesn’t mean death – it means the PCs lost more than 20% of their resources. In this light, the “getting twice as many successes as you’re allowed failures” of the 4E skill tests makes sense.

 

Let us say then that each “monster” requires 2 successes before you have 1 failure. So a test of 1 “monster” would require making 2 rolls with and fails if you miss a single roll. A test with 4 “monsters” would require 8 successes before you get 4 failures. A test with 12 “Monsters” would require 24 successes prior to 12 failures.  I’m perfectly happy to keep this, at least until I can get some statistics on the real odds of such tests – if it turns out that requiring such strings of successes diminishes the PCs chance of success too much I’ll have to reassess.

 

That’s one part out of the way…now on to setting the Challenge Ratings to Difficulty Levels. Here, 4^th edition is very clear: your % chance of success on an at-level roll is 35%, +/- 10 based on your attribute. For level -1 it’s 55% +/-10, for level +2 its 15% +/-10. If it is a skill based and you don’t have the skill, drop those by 25%. Above that difficulty and you shouldn’t be attempting it; below that and you don’t have to roll. As I said, that strikes me as being very narrow. I have my own ideas on this in a big spreadsheet that calculates the approximate final modifiers for PC skills by level based on which if 5 importance levels they ascribe to the skill and which of 3 importance levels they ascribe to the underlying attribute. But I’ll wait on sharing that until after I can figure out the overall success chance for a Skill Challenge. And that requires an answer to the question at the top of the post.

Comments

[mneme]

Are you using the pre-eratta numbers or the pre-eratta numbers?

The pre-eratta numbers were totally fscked. You had about a 50% of success per roll, but you needed to roll more than twice as many successes as failures. The result was that without a lot of cheese (aid another, way-boosted abilities) you had very low chances of successes at any level, and had almost certain chances of success if you could do aid another endlessly.

The post-eratta version is better (ie, the PCs will usually pass), but has its own problems, mostly in that it's too easy (as they largely solved the problem by dropping the difficulty numbers, though they also, IIRC, fixed "failures" at 3, so that higher-complexity problems are actually harder.

Note, btw, that since Aid Another is a possiblity, not all PCs have to make a roll on their turn; they can try to Aid instead! Or you could have a skill challenge running on the same rounds as a combat, with PCs choosing to either make a combat action or make a roll on the skill challenge.

If you poke about on Enworld, there are some excellent numerical analyses, not only of the first and post-eratta Wizards versions, but also of several alternative ideas that try to solve some of the same problems.

I don't know whether any of these systems included it, but my first thought seeing the Wizards system was that you really needed deadlines or NPC making rolls and shooting for failures. That way, you can have all the Aid Another business you want, and you still get tactical complexity as PCs only have a certain amount of time to play the challenge game before they automatically lose, so they need to make their turns count.

Of course, you know, you could just run 4E by the rules, and throw challenges (skill or otherwise) at PCs that deviated by +/- 4 from their own character level. I don't see how that's even deviating from the RAW, though lower level challenges are, of course, worth fewer XP, as they're not at-level challenges (and the same for higher-level challenges).

[brianrogers.livejournal.com]

Where are the post eratta numbers? I'm drawing from the first printing of the DMG. And it's a little sad that their first printing contained such a glaring error.

I'll look over at the enworld stuff, but I really do want to see a formula for it so I can continue my own reverse engineering.

[mneme]

There were definatley formulae in the enworld numbers.

[brianrogers.livejournal.com]

OK, I found the revisions in a PDF on the WOTC web page. I see it also removes the instruction to make it harder if the player has the skill...good to see.

[mneme]

I'm not sure where the instruction to make it harder if the player has the skill comes from -- iirc, the removal is the instruction to make skill checks harder than other checks (but weapon vs AC checks are still harder, as they get a proficiency bonus that at least cancels that out).

[brianrogers.livejournal.com]

I'll just keep replying to the same thing here. Clearly I need a drink.

Anyway, you're right - they changed things, but made them wacky in different ways. The chances are now much higher, with a 90% if it's a skill you have in your best stat, 50% for unskilled in your worst. Cutting down the number of allowable failures likely makes the longer skill challenges too challenging, as getting 9 straight successes - almost a requirement for a 12 out of 14 challenge - has a 39% chance for the 90% skilled+high stat character but drops to 13% for an 80% skilled + mid stat one. I'd rather see the number of failures allowed go up because it a) more closely approximates the length of combat and 2) draws out the test a bit to increase the potential drama.

However, I am considering altering the DC of the rolls based on the test length to keep the percent chances of overall succcess somewhat comperable. Plus, having higher DCs for the shorter tests gives them a minimum bar - the best in the party has to tbe the one to do this short test, while the longer tests can get more people involved.

I appreciate the limiting of Aid Another to one or two per roll to get everyone involved, but how about just allowing the non-skilled people to sit the test out and let the specialist shine? Why is my taciturn, sullen dwarvish fighter even talking to the Baron?

At least they stopped advocating that finding clever ways to play to your strengths should net you no bonus mathmatically. That was truly irksome.

[notthebuddha.livejournal.com]

I did a spreadsheet where the probability of zero sucesses is 1 after zero trials, and each line is another trial, with the chance of staying the same is multiplied by 0.4 and the probability of adding one success is multiplied by 0.6. I get 88.1% for 8 or more successes after 11 trials.

Beating 4 monsters that each require two successes before 1 failure would be more like 8 successes and no failures, for an 8.96% chance. Unless there is some way to soak up failures?