D&D 4E Showing the Math: Proving that 4e’s Skill Challenge system is broken (math heavy)

[rabindranath72]

Dude, I just taught a course on this stuff last semester. You can perfectly model this situation with a binomial random variable exactly as he did. Roll 5 dice at once and see if you get 4 or 5 successes is type 1. If you were to make a tree diagram and continue branches where the experiment technically stops, you'll notice that only the first 5 twigs matter.

It is a distribution of waiting times, quite different from a binomial distribution. It is the distribution of waiting times in a Bernoulli process.

 

[Eldorian]

It is a distribution of waiting times, quite different from a binomial distribution. It is the distribution of waiting times in a Bernoulli process.

What are you, a statistician? Only a statistician can take simple probability calculations and use some oddly named model to entirely mess it up. The OP's reasoning is perfect. I'd have given him an A. In fact, his reasoning is better than yours, because yours requires you to use the fact that men before you used the OP's reasoning and created this random variable.

PS:

<--- mathematician.

 

[Mephistopheles]

What are you, a statistician? Only a statistician can take simple probability calculations and use some oddly named model to entirely mess it up. The OP's reasoning is perfect. I'd have given him an A. In fact, his reasoning is better than yours, because yours requires you to use the fact that men before you used the OP's reasoning and created this random variable.

PS:

<--- mathematician.

Them's fightin' words!

I haven't seen talk like that since my days at university. There was an incident in the Mathematics Faculty that resulted in a lost eye, a broken pen, and a ruined pocket protector.

 

[Eldorian]

Them's fightin' words!

I haven't seen talk like that since my days at university. There was an incident in the Mathematics Faculty that resulted in a lost eye, a broken pen, and a ruined pocket protector.

Hey, at my university, those damned statisticians used to have their offices in our building. Luckily, we kicked them out before I really became entrenched here.

 

[pemerton]

My maths has been weakened by the passage of years, but assuming a 50/50 chance of success on each check, and just plotting out a probability tree for a Complexity 1 challenge, I get that the odds of success are 6/32, or a bit less than 20% (1/16 of 4 successes straight away, plus 1 chance each in 32 of having the single fail on the 1st, 2nd, 3rd or 4th roll). To bring the overall chance up to 75% is going to require quite a good deal more than a 50% chance on each check.

At 60% chance of success per check, my probability tree with a bit of handwaving to speed things up suggests 1053/3125, or somewhat over 30%, as the overall chance of success for Complexity 1.

At 80% chance of success per check, I make it 2304/3125, or a little under 75%.

An 80% chance of hitting DC 20 requires a bonus of +15 - but at that point it becomes automatic due to taking 10, doesn't it?

If taking 10 is ruled out, then we're looking at +4 from stat, +5 from training, let's say +1 from level (as the DCs are meant to apply to levels 1-3) and then a miscellaneous +5 (+3 focus, +2 circumstance or aid another?).

At level 30, an 80% chance to hit DC 34 requires a +29: let's say +6 from stat, +5 from training, +15 from level and then a miscellaneous +3 (focus, or a magic item and/or a circumstance bonus).

So I'm not sure that the numbers are wrong, just that 1st level PCs seem to be victims of their lack of level, feats and magic items.

 

[FireLance]

So I'm not sure that the numbers are wrong, just that 1st level PCs seem to be victims of their lack of level, feats and magic items.

What I'm deploring is that the lack of feats and magic items has apparently not been taken into account when determining the DCs for low-level characters.

 

[rabindranath72]

What are you, a statistician? Only a statistician can take simple probability calculations and use some oddly named model to entirely mess it up. The OP's reasoning is perfect. I'd have given him an A. In fact, his reasoning is better than yours, because yours requires you to use the fact that men before you used the OP's reasoning and created this random variable.

PS:

<--- mathematician.

Ehy, Mr. Funny, I am a mathematician.
But since you are smarter than me, you can explain this:
1) We're interested in the number of failures before reaching a fixed number of successes.
2) The experiment consists of a sequence of independent trials.
3) The probability of success, p, is constant from one trial to another.

We want the probability of x failures before a number of successes r.
This is (Polya/Pascal/negative binomial distribution):
P(X=x|r,p)=OVER(x+r-1, r-1)*p^r*(1-p)^x

For the most basic skill challenge outlined above, x=2, r=4, p=0.55
If you do the math (you are a mathematician, right?), then you get something like:
P(X=2|4,0.55)=0.1240

So, there is only a 12% chance of 2 failures before 4 successes.

Clearly, this result is quite different from what the OP got.

Now, since you are the smarter here, you can either disprove me, or the other guy. I am fine in either case.

Thanks,

EDIT: Oh, obviously one might want the cumulative probability, so:
P(X<=2|4, 0.55)=P(X=0)+P(X=1)+P(X=2)=0.2552

So, the probability of getting AT MOST 2 failures is 0.26

 

[vagabundo]

You know, I knew there was a reason I hated probability and this thread is IT!!!

Anyway, dont WotC have a full time maths guy working for them? If so did skill challenges slip through or do we need some erratum??

Two points from the OP:

- At 8th (party) level we see the complexity do a flip, and more complex challenges become easier, for 1st level skill challenge. Is this because of the level disparity: does the same thing happen for a 10th level party for a 3rd level skill challenge? If so maybe PCs should auto-win on skill challenges 5 below their level.

- What are people best guesses at the non-broken DCs for skill challenges from the chart (DMG pg42), maybe a +2 for skill checks rather than a +5...

 

[two]

Ehy, Mr. Funny, I am a mathematician.
But since you are smarter than me, you can explain this:
1) We're interested in the number of failures before reaching a fixed number of successes.
2) The experiment consists of a sequence of independent trials.
3) The probability of success, p, is constant from one trial to another.

We want the probability of x failures before a number of successes r.
This is (Polya/Pascal/negative binomial distribution):
P(X=x|r,p)=OVER(x+r-1, r-1)*p^r*(1-p)^x

For the most basic skill challenge outlined above, x=2, r=4, p=0.55
If you do the math (you are a mathematician, right?), then you get something like:
P(X=2|4,0.55)=0.1240

So, there is only a 12% chance of 2 failures before 4 successes.

Clearly, this result is quite different from what the OP got.

Now, since you are the smarter here, you can either disprove me, or the other guy. I am fine in either case.

Thanks,

I love interdepartmental donnybrooks.

I'm on the side of pure math vs. the stats guy.

What about you?

I give pure math 3 to 1 over the Pascal variable.

Taking bets now!

 

[pemerton]

What I'm deploring is that the lack of feats and magic items has apparently not been taken into account when determining the DCs for low-level characters.

Fair enough!

Racial bonuses will presumably make a difference, though - bring on the half-elves!