Math help - statistics, probability, and skill challenges.

[RangerWickett]

Say I want to do a pseudo skill challenge, where the PCs have to navigate a fiery hallway. Each round each PC has to make an Acrobatics check to avoid taking 5 damage, and he needs 3 successes to get to the far side. If he fails he cannot progress forward.

I need help figuring out what DC to use so that I don't kill the PCs, or make it unchallenging.

Let's assume the party's 10th level, so an untrained schmo wizard will have a +5 Acrobatics check. I'll just assume that the rogue with his +17 modifier can basically ignore the threat.

So depending on what DC I set, what kind of average damage am I looking at? Like, if it were DC 15, the wizard has a 45% chance of taking 5 damage each round he makes a check, but how many rounds will he make a check, on average?

 

[Elric]

Say I want to do a pseudo skill challenge, where the PCs have to navigate a fiery hallway. Each round each PC has to make an Acrobatics check to avoid taking 5 damage, and he needs 3 successes to get to the far side. If he fails he cannot progress forward.

So depending on what DC I set, what kind of average damage am I looking at? Like, if it were DC 15, the wizard has a 45% chance of taking 5 damage each round he makes a check, but how many rounds will he make a check, on average?

Average number of trials (rounds) needed to succeed a total of N times, when you have a p chance to succeed each trial= N/p.

To get the average number of failures, just subtract off the N successes in this sequence.

Average number of failures before succeeding those N times= N/p - N = N*(1-p)/p.

In this case, damage = 5* number of failures and N=3, so Average damage taken= 15*(1-p)/p, where p is the wizard's chance to succeed each round. For example, if the wizard had a 50% chance to succeed each round, it would take him an average of 6 rounds to complete the challenge and he'd take an average of 15 damage in this time.

 

[Tenniel]

To find out more about probabilities for skill challenges look up the negative binomial distribution. This describes the distribution of the number of trials (number of rolls) needed to get a certain number of successes (number of rolls beating the DC) of an event of a given probability (DC). The previous poster has probably given the most vital piece of information, the average number of trial (i.e. avg no. of rolls).

 

[chitzk0i]

Acrobatics is the only skill to avoid the damage? I feel sorry for anybody in plate+shield in your party.

This really sounds more like a trap or hazard. I think it would work better if you made it fire damage if you move more than three squares in a round without making an acrobatics check.

 

[sfedi]

Just in case you got lost in the above (correct) formulas I'll give you a rundown of the results:

Acrobatics DC 5

Damage expected for skill bonus 5: None
Damage expected for skill bonus 10: None
Damage expected for skill bonus 15: None
Damage expected for skill bonus 17: None

Acrobatics DC 7

Damage expected for skill bonus 5: 0.789473684210526
Damage expected for skill bonus 10: None
Damage expected for skill bonus 15: None
Damage expected for skill bonus 17: None

Acrobatics DC 10

Damage expected for skill bonus 5: 3.75
Damage expected for skill bonus 10: None
Damage expected for skill bonus 15: None
Damage expected for skill bonus 17: None

Acrobatics DC 12

Damage expected for skill bonus 5: 6.42857142857143
Damage expected for skill bonus 10: 0.789473684210526
Damage expected for skill bonus 15: None
Damage expected for skill bonus 17: None

Acrobatics DC 15

Damage expected for skill bonus 5: 12.2727272727273
Damage expected for skill bonus 10: 3.75
Damage expected for skill bonus 15: None
Damage expected for skill bonus 17: None

Acrobatics DC 17

Damage expected for skill bonus 5: 18.3333333333333
Damage expected for skill bonus 10: 6.42857142857143
Damage expected for skill bonus 15: 0.789473684210526
Damage expected for skill bonus 17: None

Acrobatics DC 20

Damage expected for skill bonus 5: 35.0
Damage expected for skill bonus 10: 12.2727272727273
Damage expected for skill bonus 15: 3.75
Damage expected for skill bonus 17: 1.66666666666667

Acrobatics DC 21

Damage expected for skill bonus 5: 45.0
Damage expected for skill bonus 10: 15.0
Damage expected for skill bonus 15: 5.0
Damage expected for skill bonus 17: 2.64705882352941

Acrobatics DC 22

Damage expected for skill bonus 5: 60.0
Damage expected for skill bonus 10: 18.3333333333333
Damage expected for skill bonus 15: 6.42857142857143
Damage expected for skill bonus 17: 3.75

Acrobatics DC 23

Damage expected for skill bonus 5: 85.0
Damage expected for skill bonus 10: 22.5
Damage expected for skill bonus 15: 8.07692307692308
Damage expected for skill bonus 17: 5.0

Acrobatics DC 24

Damage expected for skill bonus 5: 135.0
Damage expected for skill bonus 10: 27.8571428571429
Damage expected for skill bonus 15: 10.0
Damage expected for skill bonus 17: 6.42857142857143

Acrobatics DC 25

Damage expected for skill bonus 5: 285.0
Damage expected for skill bonus 10: 35.0
Damage expected for skill bonus 15: 12.2727272727273
Damage expected for skill bonus 17: 8.07692307692308

Acrobatics DC 26

Damage expected for skill bonus 5: Infinite
Damage expected for skill bonus 10: 45.0
Damage expected for skill bonus 15: 15.0
Damage expected for skill bonus 17: 10.0


Note that these are the expected damage for 3 checks, each with a damage of 5.

The expected damage for each individual check should be the above divided by 3.

Please, if someone can double-check in case I messed it up, thanks in advance

 

[sfedi]

I'd like to note/add that the problem RangerWickett posted is essentially the same as ongoing damage (you take X damage until you succeed on the check/save).

Si, here we have how much expected damage has a 5 ongoing damage:

9.090909...

This is very interesting, because now we can make an equivalence between an attack with or without ongoing damage:

1d8 + 7 = 1d4 + 5 ongoing (for example)

The formula is: 1 / (1 - 0.45) * ongoingDamage

0.45 is the chance of failing the save/check.

So you can estimate how much damage a Human with those feats that grant bonuses to saves takes by changing this number.

And changing the ongoing damage allow us to put other ongoing damage values with confidence of what we are doing.

Note that ongoing damage isn't cumulative: you can't have two ongoing damage of the same type affecting you, so it should be a little less valuable.

 

[Rothe_]

I think one more approach might help out here.
You should calculate how likely is it to fail so many times as to lose all your hp.
Expected damage is fine and all that, but what interests me is the risk of actually getting killed - the expected damage does not really tell you that.

So, if you have a 10th level wizard with 50 hp, how likely is it for him to lose all of his hp before succeeding in the check? Basically he can fail 9 times. The 10th time knocks him out.

I don't remember how to calculate this.

 

[Nichwee]

Assuming the odds of failing a Roll = X. Thus odds on succeeding a roll = 1 - X.
Assuming Y successes = Completed.
Assuming Z failures = death.

Odds on death = sum of all version where Z fails occur before Y successes.
As the order of the fails/successes is irrelevent (as long as less than Y successes occur in total) we have versions where 0, 1, 2, ..., Y-1 successes occur as all the versions

Odds on death = X^Z *sum(1 + [1-X] + [1-X]^2 + .... + [1-X]^[Y-1]).
Odds on death = X^Z *sum([1-X]^N) for N = 0 to Y-1.

The [1-X]^N bits are for each case where a number of successes less than Y are rolled, where N denotes the number of successes gained for that version of the outcome.

 

[WalterKovacs]

One issue with the "odds of death" ... presumably you won't be cut off from the rest of the party, so while it's possible to be inside the trap (where you can't just "give up" and leave) you will still have access to some healing. So you'd probably have your second wind, maybe a potion or two, plus the leader's encounter healing powers to keep you alive. In those cases you may not end up dead, but the "encounter" will be a bit of a drain on resources, which is part of the goal. Then again a skill challenge with only a single skill being used is not necessarily the most exciting way of going about challenging the party.