Questions for Probability People


Advice


Here's my question:

Let's say somebody has an ability score of 12. They take 1d6 points of ability damage at night, but recover 4 points the following day. Using probability, how many days will it take before their score drops to zero?

And how about the same question, only this time with an ability score of 11?

Thanks in advance...


Disclaimer, I am not a "probability people", but I am a "modeling and sim people". In this case, the probability is a little tricky and it is far easier to just run a simple simulation.

I counted a day to mean that you survived the night (didn't drop to 0). The minimum amount of days is 4 (you would start day 4 with a 6, but die that night with 6 damage).

I ran 10 experiments of 10000 runs (of days it takes to get to Int<=0 from Int=12 starting), I got:

{min, max, avg, stddev}
{4,1386,128,122}
{4,1204,128,123}
{4,1044,129,122}
{4,1088,128,120}
{4,950,129,122}
{4,1212,129,123}
{4,1195,129,123}
{4,1010,129,122}
{4,1455,127,120}
{4,1101,127,119}

You can interpret these results however you want, but there are some patterns you can see

(1) The minimum amount of days is 4 (as noted above)
(2) The max can vary significantly depending on the rolls
(3) The average is going to be around 128 days
(4) The 1st stddev is around 122 days

I repeated for a starting score of 11 (only 5 trials this time), here are the data sets
{min, max, avg, stddev}
{4,982,129,119}
{4,1265,128,121}
{4,1387,127,119}
{4,1103,128,120}
{4,1090,127,121}

Enjoy!


Here is my Matlab code for anyone interested(ignore random typos as I had to re-transcribe this, not copy/paste).

Runs = zeros(10000,1);
for Index = 1:10000
Int=12;
Days=0;
while (Int>0)
Int=Int+4;
if Int > 12
Int=12;
end
dmg=randi(6,1);
Int=Int-dmg;
Days=Days+1;
end
Runs(index)=Days;
end
AvgDays = mean(Runs(:,1));
StdDev = std(Runs(:,1));
MinR = min (Runs(:,1));
MaxR = max(Runs(:,1));
disp([MinR MaxR AvgDays StdDev]);


edit:ninja'd

I won't claim to be an expert, but examining the problem it seems like your best bet is to run a simulation to approximate the answer. I threw together a spreadsheet with 50 trials and all of them managed to get to 12 before 1000 rolls.
Mean: 126.32
Min: 4
Max: 506

For 11...
Mean: 99.86
Min: 5
Max: 661

edit 2: justaworm, obviously you did way more trials so your results are more probable, but I'm surprised that your averages for 11 were so close to the ones for 12.

Grand Lodge

GroovyBoy wrote:

Here's my question:

Let's say somebody has an ability score of 12. They take 1d6 points of ability damage at night, but recover 4 points the following day. Using probability, how many days will it take before their score drops to zero?

And how about the same question, only this time with an ability score of 11?

Thanks in advance...

Tell us what disease you caught, and how far from civilization you are...


It is intensely hard to die from this in any reasonable amount of time. You have to roll consistently high several days in a row in order for it to happen. You only have a 1 in 3 chance of actually progressing towards the kill point each night--even rolling a 4 ends up breaking even.

I am curious, though, how are you getting 4 back per day?

Grand Lodge

mplindustries wrote:

It is intensely hard to die from this in any reasonable amount of time. You have to roll consistently high several days in a row in order for it to happen. You only have a 1 in 3 chance of actually progressing towards the kill point each night--even rolling a 4 ends up breaking even.

I am curious, though, how are you getting 4 back per day?

Likely a Maximized Restoration from the party's Oracle who hasn't learned Remove Disease.

It's possible, especially if it's an oracle specced for Melee, they only have so many spells known. Restoration spells address a symptom (ability damage) that can have multiple causes (poison, disease, special monster attacks, etc. Rather than having something to negate every possible cause, they invested in something mitigate the effect.

Basically, he's got a bandaid when he needs a surgeon.

The party should have invested in a potion or scroll though.


It's basically an exponential decay, with a half-life of around 148 days. So, you're ~50% likely to hit 0 before 148 days, ~75% likely before 296 days, etc. Here's some graphs, where the upper right plot shows your probability of hitting 0 on any given day in percent (blue is simulated, pink is an exponential fit) and the lower right plot is the logarithm of that probability (again, blue is simulated, pink is the fit.)

R code:
a_0 = 12
time = 365
runs = 100000

z = rep(0,time)

for (r in 1:runs) {

a = a_0

score = rep(0,time)

t = 0

while ((t<time)&(a>4)) {
d = ceiling(runif(1,0,1)*6)
a = min(a_0,a-d+4)
score[t] = a
t = t + 1
if (a==4) {z[t] = z[t] + 1}
}

}


If you can't find your way back to civilization in a few months you likely have bigger problems than the ability damage. I wonder how bad it is for the poor character who dumped to a 7. Dumping to a 5 is going to get a character killed with at least a 1 in 3 chance of dying each night. If you had a 15 or 16 you could probably survive quite a while.

Grand Lodge

I've had to GM for characters whose constitution dips below 10. It's godawful for everyone, including me, who has to emulate the universe violently rejecting the idea of an armorless changeling cleric claw their enemies with 12 STR and 8 CON.


Remembering my old statistics classes... charting this looks like a Poisson Lambda Distribution with like 2-degrees of freedom. It peaks somewhere around 15-20 days, meaning that is most likely.


Torchlyte wrote:


edit 2: justaworm, obviously you did way more trials so your results are more probable, but I'm surprised that your averages for 11 were so close to the ones for 12.

Since you mentioned it, I was too. I found the problem. Good catch!

When I ran with 11, I only changed the starting value, and not my checks, haha. Here is better data for Stat=11:

{4,746,86,80}
{4,819,85,79}
{4,710,84,79}
{4,704,85,80}
{4,920,86,80}

Change the damage done also has a much more profound affect in this experiment. Just going to a d8 will usually have you dead within 2 weeks (stat=12).

Grand Lodge

Adventure Path Charter Subscriber; Pathfinder Starfinder Adventure Path Subscriber
justaworm wrote:
Disclaimer, I am not a "probability people", but I am a "modeling and sim people". In this case, the probability is a little tricky and it is far easier to just run a simple simulation.

You have too much time on your hands....

-Skeld


What ... it took like 5 minutes. :)


Those "mean" numbers do not mean to much with a skewed distribution like this. You are better-off looking at the "mode" numbers -- those that are most likely.


A better gauge for your chances of survival is the median. "Typically," you can expect to survive for around 90 nights!

In general, starting with 12 ability points, the probabilities of your death within x days are:

death within 1000 days: 99.97%
death within 500 days: 98.23%
death within 200 days: 79.46%
death within 100 days: 53.34%
death within 50 days: 29.66%
death within 20 days: 10.02%
death within 10 days: 2.69%
death within 5 days: 0.08%

Figures (including median) are based on a 1,000,000 iterations of the simulation.

GroovyBoy wrote:

Here's my question:

Let's say somebody has an ability score of 12. They take 1d6 points of ability damage at night, but recover 4 points the following day. Using probability, how many days will it take before their score drops to zero?

And how about the same question, only this time with an ability score of 11?

Thanks in advance...


I just ran my chart again with 100,000 trials, and it definitely peaks at 15-16.


And for a starting ability score of 11, your median life expectancy drops to 60 nights.

GroovyBoy wrote:

Here's my question:

Let's say somebody has an ability score of 12. They take 1d6 points of ability damage at night, but recover 4 points the following day. Using probability, how many days will it take before their score drops to zero?

And how about the same question, only this time with an ability score of 11?

Thanks in advance...


Consider average rolls:
Average of a d6 is 3.5, so you are most likely to roll a 3 or 4 on average. Which means on average you heal aan equal amount or more than you take. In fact, only on rolls of 5 or 6 do you decrease.

A problem that I am wondering about and whether the math above (that I ignored) has taken into account is the possibility that while you could roll a 5 one night, and a 3 the next night. With the maximized restoration occuring both nights you're at effectively no penalty again.

You can really go for a long time without needing to worry about this too much. You would have to consistently roll 5s or 6s for it to be a problem, because otherwise the restoration will heal up latent ability damage from other nights.


Heh. The answer to this is fight for 2-10 more days, level and cure it!

I'm currently in Rappan Athuk and our party has 5 members with incurable magic diseases that debilitate hp/hour and abilities like con /day. We're level 4 and there is zero civilization within 100's of miles. We're pouring all of our Druid and Cleric spells into surviving until 5. Stupid caster level checks!

Once more unto the breach!


I don't think it makes sense to say "only the mode is important," unless you are wagering on which day you are most likely to die. The mean or median may be interesting to calculate, but the REAL thing that matters is the full distribution. And since it's an exponential, the half-life is probably the most interesting (and that will be reasonably close to the median.)


I think it would be really cool to have a character who actually died that way. But maybe I'm alone in this.


I made a chart here for the 12-hp one. The cursor is pointing at the median.


Brf wrote:
Remembering my old statistics classes... charting this looks like a Poisson Lambda Distribution with like 2-degrees of freedom. It peaks somewhere around 15-20 days, meaning that is most likely.

I disagree. The Poisson Distribution has to do with the probability of a given number of specific incidents occurring over a given time period. (Such as how many cars pas through a given intersection in a day, or how many shoppers come to a given mall during a given day.)

http://www.umass.edu/wsp/resources/poisson/#application


I said it looks like the same shape distribution, not that it was the same application.


RumpinRufus wrote:
I don't think it makes sense to say "only the mode is important," unless you are wagering on which day you are most likely to die. The mean or median may be interesting to calculate, but the REAL thing that matters is the full distribution. And since it's an exponential, the half-life is probably the most interesting (and that will be reasonably close to the median.)

Agree. I fail to see how the mode is the most interesting here. Especially since the mode can vary quite a bit while the mean and stddev stays pretty stable.


Yup. Looking at the chart, the mode is more like the likely-minimum. The median is more down in the focus.


Well, the simulated mode may vary, but the analytical mode will just be one number. The mode varies more in a simulation because it's established after just a few time steps, while the mean and stddev are averaged over the entire data set.

The mean and stddev have their own set of problems though, mostly that they tell you nothing about the shape of the distribution. A normal distribution could have the same mean and stddev as this distribution, but the chance of dying on any given day would be much different if you assumed it were a normal distribution instead of an exponential decay.


Right. The Mean and Stddev are really only meaningful in a standard/symmetrical distribution, not one like this one. The Median still works well though. Did you see my graph?


Yeah, I saw your graph. It looks like my graph ;)


Interesting, thanks. I can't go to linked graphs, but I will take a look at home.


Mythic Evil Lincoln wrote:
I think it would be really cool to have a character who actually died that way. But maybe I'm alone in this.

It certainly would be novel. I would give it a thumbs up.

I would prefer it be a random countdown timer to your death that you could potentially stop.

Designer

1 person marked this as a favorite.
JJ Jordan wrote:
Mythic Evil Lincoln wrote:
I think it would be really cool to have a character who actually died that way. But maybe I'm alone in this.

It certainly would be novel. I would give it a thumbs up.

I would prefer it be a random countdown timer to your death that you could potentially stop.

"The instant this random number generator reaches zero, you'll be executed...Ten. Three. Twelve. Three again."


"I do not know why this simulation of 2d6 is not giving me a zero!"


Claxon wrote:

Consider average rolls:

Average of a d6 is 3.5, so you are most likely to roll a 3 or 4 on average. Which means on average you heal aan equal amount or more than you take. In fact, only on rolls of 5 or 6 do you decrease.

A problem that I am wondering about and whether the math above (that I ignored) has taken into account is the possibility that while you could roll a 5 one night, and a 3 the next night. With the maximized restoration occuring both nights you're at effectively no penalty again.

You can really go for a long time without needing to worry about this too much. You would have to consistently roll 5s or 6s for it to be a problem, because otherwise the restoration will heal up latent ability damage from other nights.

Yeah, there is a 1/3 chance of your condition worsening, a 1/6 chance of your condition getting a slight improvement (potentially canceling out the previous drop in condition) and a 1/6 chance of your condition absolutely improving.

Overall, this is not that bad. There is a reason why a lot of means I've seen quoted at a glance appear to be over 3 months.


lemeres wrote:
Claxon wrote:

Consider average rolls:

Average of a d6 is 3.5, so you are most likely to roll a 3 or 4 on average. Which means on average you heal aan equal amount or more than you take. In fact, only on rolls of 5 or 6 do you decrease.

A problem that I am wondering about and whether the math above (that I ignored) has taken into account is the possibility that while you could roll a 5 one night, and a 3 the next night. With the maximized restoration occuring both nights you're at effectively no penalty again.

You can really go for a long time without needing to worry about this too much. You would have to consistently roll 5s or 6s for it to be a problem, because otherwise the restoration will heal up latent ability damage from other nights.

Yeah, there is a 1/3 chance of your condition worsening, a 1/6 chance of your condition getting a slight improvement (potentially canceling out the previous drop in condition) and a 1/6 chance of your condition absolutely improving.

Overall, this is not that bad. There is a reason why a lot of means I've seen quoted at a glance appear to be over 3 months.

Indeed, your chance to get worse is equal to your chance to get better. You really have to roll successive 5s and 6s to die. It would happen if you ignore the condition or are unable to fix it for an extended peiord of time. But this seems...unlikely. Or at least a contrived setup to make it so.

And if such was the case I would mostly be annoyed.


Inevitables show up. They simulate 1000 combats with the party. They say, "No thanks." Then they just leave.

I would like to see the original poster give us some background for this question. I can see it as a great mechanic to push characters through a plot and minimize the 15-minute adventuring day effect.


First, I would like to thank everybody who did all that math I totally didn't understand.

Second, for the curious, some background info: I need this data to flesh out the background of an adventure where one NPC has already died of a mysterious disease, and a second is in the middle of its throes. Basically, they are being visited every night by a supernatural entity who is draining their life essence (Con damage), but during the day are being treated by a doctor as best he can.

Except that those average times (128 and 100 days) are a bit too long for what I'm looking for. Could anybody do the same calculations again if the people were only getting 2 points back a day? Pretty please?


Claxon wrote:

Indeed, your chance to get worse is equal to your chance to get better. You really have to roll successive 5s and 6s to die. It would happen if you ignore the condition or are unable to fix it for an extended peiord of time. But this seems...unlikely. Or at least a contrived setup to make it so.

And if such was the case I would mostly be annoyed.

Heck, your chances of getting better are larger. 4 is neutral, 6 and 2 cancel eachother out, 3 and 5 cancel eachother out, and that leaves 1 which allows you to recover 25% of your constitution. 3 numbers heal, 2 numbers hurt.

Of course, just because it is unlikely that you will die doesn't mean that this isn't a rather big problem. You are still going to usually have at least a few points of CON damage most of the time. So that means maybe a loss of 1-2 fort and 1-2 hp per hit dice (so 1-40 HP loss). You are stick coughing up a lung every couple of days here.


GroovyBoy wrote:
Could anybody do the same calculations again if the people were only getting 2 points back a day? Pretty please?

That changes a lot, it becomes very likely you'll die in less than a week (it's more than 60% likely you'll die between 3 and 7 days) and extremely unlikely that you'll live longer than two weeks.


Okay, how about 1d4 damage, get back 2 a day?


RumpinRufus wrote:
GroovyBoy wrote:
Could anybody do the same calculations again if the people were only getting 2 points back a day? Pretty please?
That changes a lot, it becomes very likely you'll die in less than a week (it's more than 50% likely you'll die between 3 and 7 days) and extremely unlikely that you'll live longer than two weeks.

Yeah, 2/3 hurt, 1/6 is neutral, and 1/6 slightly helps in that case.

Personally, if I was designing it for a somewhat slow death, then 2 points versus 1d4 ability score damage might work better. You are less likely to have some sudden big loss, and it is still tipped towards the 'dying' side. 1d6 vs 3 might be similar, but still gives painful dips (which might be what the GM is looking for, I suppose)

edit: darn it, ninja'd. Eh, I guess it was kind of a fairly straight forward progression for dialing the damage/healing ratio after all this 3 months stuff.


GroovyBoy wrote:
Okay, how about 1d4 damage, get back 2 a day?

That gives you about two and half weeks.


I can work with that...thanks!

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