This is how much damage a character takes from various poisons on average (MATH INVOLVED)


Pathfinder First Edition General Discussion


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This would be more fancy if I could post images here, but we'll have to do without that. I've tracked down an interesting formula, and with some manipulations it can be quite useful.

What I have is a formula to approximate the number of rolls you have to do, on average, until you will succeed in negating a poison/disease that requires two successful saves to end.

The formula looks as follows

1 / p^2 + 1 / p

Where p is the probably to succeed in a single roll.

Note that this formula does NOT correctly adjust for the possibility that failing a save would result in lowering the save bonus itself as a result. For example a poison that causes constitution damage would lower your fortitude save, thereby making your future saves even less likely to happen. This is not taken into account because it makes it even more complex to calculate than it already is :P

First: If we insert p = 1 in the formula we get 1/1^2 + 1/1 which is 2. When you're 100% likely to succeed every save, at WORST you will have to make two saves to escape a poison.

If we insert p = 0.5 we get 6. So it'd take on average something like 6 rolls until you succeed.

We can calculate p for any given save with the following formula

(20+s-d)/20

Where s is your save bonus and d is the DC. Example: A character with a constitution save of 5 rolls a save against a poison with DC 19.

(20+5-19)/20 = 0.3 (30%)

If we plug in this in the initial equation we get 14.44, which is really quite bad... It'd take an average of 14 rounds to break out of this poison.

Note, of course, that we can calculate the amount of damage taken on average too. I've made a table of average damage per die below:

1d3 = 2
1d4 = 2.5
1d6 = 3.5

With another formula we can proceed to calculate how much ability damage you will take on average. It looks like this

d = e(1-p)(k-2)

Where p is the probability to successfully save against the poison, e is the average damage value from the above table, and k is the number of rolls from above.

Example: k value = 14.44, p = 0.3, and 3.5 as we use a d6.
3.5 * (1-0.3) * (14.44-2) = 30.475

MASSIVE DAMAGE

I've put together a complete formula here that you can paste into wolframalpha or any other calculator:

damage = e(1-((20+s-d)/20))((1 / ((20+s-d)/20)^2 + 1 / ((20+s-d)/20))-2)

e = damage value from the above table
s = relevant save bonus
d = poison DC

In short: This gives you approximation of how much damage a character would take if afflicted by a poison with save DC d, expected die damage e (from the table) and save bonus s.

I will revisit this later, maybe next weekend, with a more rigorous examination of the probabilities. There might be mistakes in what I have here, but they shouldn't be that severe. If you find anything, tell me.


Dotting this for future reference.


This would be useful for a DM who's writing his own encounter, to know exactly how deadly his poison might be towards a given party level.
Adjust for how much warning they might have in advance, chance to prepare or recover, etc.

I could put this in an excel sheet for my tools, nice one :)


If you're using this to estimate the lethality of a poison for an encounter, it's also important to note how much likely exposure characters will have to it. A poison effect that can get applied more than once to a character (such as on a creature's attack) will skew this table. Each time a character with an ongoing poison effect takes a successive dose of the poison, it stacks to increase duration by 1/2 its amount and increase the save DC by +2.


Nice calculations! Not a math-wizz myself but I might use it in the future for working out the cost/benefit ratio of the various poisons if I ever play a poison-using character again.
I love poisons but the traditional prohibitive cost, low chance of success, non-scaling saves, limited use (whoo... Dex-damage to a caster!) and long time to effect always prevented me from using them.


Great work like I said earlier!

Just a few things I've noticed about this formula while playing around with it in Excel

1) It does not take natural 20s or 1s into account.
2) A poison with save 20 results in a divide by 0.
3) It does not take frequency into account.


Thanks for your replies, people! I kinda forgot about the thread :P

WRoy: Are you talking about applying multiple doses of the poison on the same person? Yes that's not covered at all by this calculation.

The Quite-big-but-not-BIG Bad:
1: That is true. It might be fixable in the final equation. I cannot, however, fix that for the very first formula: 1 / p^2 + 1 / p because this is derived from another formula of rather nightmarish complexity. I'll look into it.
2: What? lol. Let me check.
3: Can you elaborate on this?


Aha. I see. I will modify the formula, until then note that when the following equation holds true, you get a division by zero:
20 - DC + save = 0

That's because in that particular case, the probability of succeeding is exactly zero, and that means the expected value is infinity :P

Silver Crusade

Ganryu wrote:

Aha. I see. I will modify the formula, until then note that when the following equation holds true, you get a division by zero:

20 - DC + save = 0

That's because in that particular case, the probability of succeeding is exactly zero, and that means the expected value is infinity :P

You always have at least a 5% chance to succeed at a saving throw because a natural 20 is always a success.


Yes I know. I'll update with a new version.

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