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View Full Version : [4e] Death Saving Throws - A question for the math types!

Saph
2009-01-26, 02:06 PM
I was looking at save bonuses in 4e and came to the interesting realisation that, while a +1 or +2 to saves doesn't help very much against save-ends effects, it makes a very big difference when you're left alone and dying. Assuming no-one's going to come and rescue you, and assuming the unlucky PC has a healing surge left, a normal character will live if he rolls a 20 on his save, and will die if he rolls 1-9 on his saving rolls three times before rolling that 20.

On the other hand, a character with a +1 save bonus, say from Human Perserverance, will live if he rolls a 19-20 before rolling three 1-8s - which is actually reasonably likely. I tried to calculate the exact chances, but unfortunately my probability skills weren't quite good enough. :)

So my question for the math-inclined on this forum is: for each net save bonus (+0, +1, +2, etc.) what are the chances of living or dying if left to make death saving throws?

And, on a related note, what are the best ways to boost saves?

- Saph

Totally Guy
2009-01-26, 02:44 PM
Ok.

Probalility of failure in a round = 9/10.
Probability of success = 1/10
This is a ratio life vs failure. Other results happen 1/2 the time but as a repeat iteration so we can ignore this.

I want to find P(Death)

P(Death given 2 failures) = P(2 failures given death)*P(Death)/P(2 failures)

P(Death given 2 failures) = 9/10

P(2 failures) = 81/100

P(2 failures given death) = Certain, 1.

So we have

9/10 = 100/81 * P (Death)

P (Death) = 729/1000

P (Life) = 1 - P (Death) = 231/1000.

For no saving modifier.

For modifier +1 we still have the same 10 "reiterate" results so really it's 2/10 life ~ 8/10 failure.

Same as before:

8/10 = 100/64 * P (Death)

P (Death) = 512/1000
P (Life) = 488/1000

For Modifier +2

7/10 = 100/49 * P (Death)

P(Death) = 343/1000
P(Life) = 657/1000

I hope I've got that right. Seems your hypothesis was right.

PinkysBrain
2009-01-26, 02:57 PM
I don't know for certain that this line of reasoning is correct glug (I've seen the "rerolls don't matter" argument before, I just never worked it through in my head). I am however fairly certain that you made the question of "what are the odds of 3 successive failures" a lot harder than it has to be :)

It's simply (9/10)^3 ...

Also (assuming rerolls don't matter) with the +1 save modifier the chances are 9/11 and 2/11, not 8/10 and 2/10.

Totally Guy
2009-01-26, 03:01 PM
It's simply (9/10)^3 ...

Oh yeah, so it is. Didn't notice that.

But I got to use Bayes theorem which is a feat I don't get to use very often.

As for the "re-rolls don't matter" it's a bit like saying:

I have no D6 so I'm going to roll a D8 until I get a result less than 7.

Eventually I'd get a number between 1 and 6 with equal odds. Which is what a D6 does.

Yakk
2009-01-26, 03:02 PM
As time is not a factor, rounds in which you roll a 10 to 19 can be ignored.

This happens to be a range of 10, leaving 10 results (out of 20) that matter.

To die, you must roll 3 deaths in a row, without rolling a not-death. Your chance of dieing is thus [(9 - B)/10]^3, where B is your saving throw bonus.

Thus, Death Chance = [729 - 243B + 27B^2 - B^3]/1000
which is actually pretty annoying to work with. :-) Note that using the original calculation of (.9 - .1B)^3 is easier!

Your chance of rolling 3 "I'm dieing" before 1 "I'm perfectly OK" is:
72.9% at +0
51.2% at +1
34.3% at +2
21.6% at +3
12.5% at +4
6.4% at +5
2.7% at +6
0.8% at +7
0.1% at +8
0.0% at +9

Saph
2009-01-26, 03:07 PM
Heh. Yeah, that's much easier. :)

Okay, so the numbers would be:

+0 saving throw bonus: 27% chance of living
+1 saving throw bonus: 49% chance of living
+2 saving throw bonus: 66% chance of living
+3 saving throw bonus: 78% chance of living
+4 saving throw bonus: 88% chance of living
+5 saving throw bonus: 94% chance of living

Wow. Even a small increase really makes them go up fast!

Trouble is, I'm not sure if it's possible to get +5 to saves - in fact, I'm not sure it's even possible to get +4. The only permanent save-booster I know is the Human Perserverance feat.

- Saph

Totally Guy
2009-01-26, 03:16 PM
The only permanent save-booster I know is the Human Perserverance feat.

- Saph

I'm going to hazard a guess that if WotC knows these numbers then there probably won't be a way to boost it further. At least not a way that stacks with human perseverance.

Kurald Galain
2009-01-26, 03:19 PM
I'm going to hazard a guess that if WotC knows these numbers then there probably won't be a way to boost it further. At least not a way that stacks with human perseverance.

False. The feat "Stubborn Survivor" (from the Forgotten Realms guide) gives a +2 bonus that stacks with Perseverance (while you're out of action points, which should be during every combat anyway because the point of AP is to use them)

RTGoodman
2009-01-26, 03:23 PM
Trouble is, I'm not sure if it's possible to get +5 to saves - in fact, I'm not sure it's even possible to get +4. The only permanent save-booster I know is the Human Perserverance feat.

I'm pretty sure there are probably some Leader powers (or maybe even Paladin powers) out there that give bonuses on saving throws. They probably give power bonuses, so they stack with the feat bonus from Human Perseverance, meaning you can probably get at least a +4 situationally.

EDIT: WAIT. Do bonuses to saving throws in general count for Death Saving Throws, too? Or do you have to have things that SPECIFICALLY give bonuses to saving throws? I wonder because of this:

Unconquered Standard of Arkhosia - Paragon Level
[...]
Property: You and allies within range gain a +2 bonus to AC and saving throws when bloodied.
Property: You and allies within range add their Constitution modifiers to their healing surge values. (Dragonborn instead add an additional +2 to their healing surge values.)
Property: You and allies within range gain a +1 bonus to death saving throws.
Property: You and allies within range gain resist 5 fire.
[...]

If general saving throw bonuses work for Death Saving Throws, then there's almost NO way to die from failing them because of the number of things you can use to get bonuses to saving throws (Human Perseverance, item bonuses, power bonuses from warlord/cleric/paladin powers, etc.). If not, it's considerably harder to get bonuses to JUST death saving throws, I think.

Artanis
2009-01-26, 03:27 PM
Wait, it's "3 in a row"? I thought it was "3 bad ones before you get a 20".

And I spent all this time working on the formula for trying to figure out the latter :smallfrown:

And no, probability is not my strong suit. It actually uses numbers :smallredface:

RTGoodman
2009-01-26, 03:32 PM
Wait, it's "3 in a row"? I thought it was "3 bad ones before you get a 20".

It's 3 failures BEFORE YOU REST, as PHB 295 indicates.

Saph
2009-01-26, 03:33 PM
False. The feat "Stubborn Survivor" (from the Forgotten Realms guide) gives a +2 bonus that stacks with Perseverance (while you're out of action points, which should be during every combat anyway because the point of AP is to use them)

Ah, that's a good one.

The other one I was thinking about was Bloodthread armour. It gives a +2 to saves while you're bloodied, and anyone below half HP is bloodied - which includes characters who are dying.

The main problem with it is that it's cloth-only, but that would give you a +5 to death saves if you stacked them all together.

RtG: I was really looking for ones that worked regardless of powers - after all, if there's someone around to use a power on you, they can just heal you up instead. I was curious if there was some way to make a character who has a decent chance of surviving being killed and left there. Seems there is.

- Saph

Yakk
2009-01-26, 03:41 PM
No, it isn't 3 in a row.

The trick is that rolls of 10 to 19 do not change the state of the situation, other than time passing. And we don't care about time passing.

Ie, if you roll a 10 to 19, things looked exactly the same. So we neglect that possibility -- if you roll 10 to 19, you just reroll.

Ie: Roll d20. On a 1 to 9, gain a death token. On a 20+, get up. On a 10 to 19, reroll.

After you evaluate all of the rerolls, this works out to be the same as rolling a d10, on a 1-9 gain a death token, on a 10+ get up.

As it happens, each +1 to the d20 with rerolls changes the probability of the "no reroll" by a +1 on the d10 roll. (This isn't true in general cases like this, but happens to be true in this case).

Imagine we had a situation where there are 3 possibilities -- A B and C. P(A)+P(B)+P(C) = 1.

However, if event B happens, we simply repeat the trial.

What is the probability we get an A before we get a C? Call this probability X.

Then X = P(A)*1 + P(B)*X + P(C)*0
or, X(1-P(B)) = P(A)
X = P(A)/(1-P(B))

If A is "get a death token", B is "don't get better or worse" and C is "get up", we end up with:
X = (.45 - bonus*.05)/.5
X = .9 - bonus *.1
as our chance of getting one "get a death token" before you roll "get up".

In order to die, you need to end up with 3 "get a death tokens" before you "get up". Once you get up, the death-spiral game ends.

Thus the chance that you get 3 "death tokens" before one "get up" is X^3, or (.9 - bonus * .1)^3, as I mentioned earlier.

(Yes, this is a slick probability trick, in order to make the situation much easier.)

RTGoodman
2009-01-26, 03:41 PM
RtG: I was really looking for ones that worked regardless of powers - after all, if there's someone around to use a power on you, they can just heal you up instead. I was curious if there was some way to make a character who has a decent chance of surviving being killed and left there. Seems there is.

Ah, right. There ARE, as it turns out, still quite a few things you can use.

-The Potion of Spirit in AV gives you a power bonus on death saving throws until the end of the encounter, so if you chug one before you go down you can get a power bonus even if you're alone.

-Several items (especially armors and neck-slot items) give item bonuses to death saving throws, ranging from +2 on up.

-Verve armor (for scale and plate) gives an UNNAMED +2 bonus to death saving throws.

-At least two paragon paths give bonuses on saves vs. death - Flamebrow Commander (for Dragonborn Warlords) lets you add your Charisma to them (read: GREAT), and Raven Herald (for Raven Queen-worshiping Rogues) gives a +4 unnamed bonus.

Using a combination of those, you should be able to make at least a couple of characters (a Human Rogue/Raven Herald with Human Perseverance and a Dragonborn Inspiring Warlord/Flamebrow Commander) who would NEVER fail a death saving throw.

Artanis
2009-01-26, 03:42 PM
It's 3 failures BEFORE YOU REST, as PHB 295 indicates.
See, that's what I thought, for exactly that reason. But people kept saying "3 in a row"...

Draz74
2009-01-26, 04:53 PM
See, that's what I thought, for exactly that reason. But people kept saying "3 in a row"...

Right, because they had already read & accepted Yakk's argument for why we're ignoring the 10-19 results. With that proviso, it is "3 in a row."

NecroRebel
2009-01-26, 06:16 PM
Using a combination of those, you should be able to make at least a couple of characters (a Human Rogue/Raven Herald with Human Perseverance and a Dragonborn Inspiring Warlord/Flamebrow Commander) who would NEVER fail a death saving throw.

Or you can use this clever technique I call "being a Warforged." This technique consists of being a Warforged and taking advantage of their Living Construct subtype which, among other things, gives you this neat little feature:
When you roll a death save, you can take the better of
your die roll or 10 as the result. You still die at your negative
hit point total.Which, clearly, makes it so that unless you have a penalty to saving throws you never fail a death saving throw.
Dragon 364's Warforged article has the same wording, but adds some words to the first line of the ability that don't actually affect the mechanics of it.

Fun, isn't it? :smallbiggrin:

Saph
2009-01-26, 06:29 PM
Or you can use this clever technique I call "being a Warforged." This technique consists of being a Warforged and taking advantage of their Living Construct subtype which, among other things, gives you this neat little feature:Which, clearly, makes it so that unless you have a penalty to saving throws you never fail a death saving throw.

Ah, but that also means you can't get a 20 and come back from death to consciousness. :)

I went back to Adventurer's Vault and found a new item - the Survivor's Belt. It's a level 11 item that lets you roll 2 death saving throws for each one, taking the higher result. Pair that up with a +2 or so in save modifiers and you should be able to reliably get a 20 on death saves, given long enough (the math is kind of ugly, but think the Survivor's Belt with a +2 to saves should give you a recovery probability in the 90%-95% range).

This means that as long as an enemy doesn't keep attacking you while you're down or hit you with a coup-de-grace, you can expect to recover from the dying condition fairly consistently. Not a bad ability to have.

- Saph

Artanis
2009-01-26, 06:32 PM
Ah, but that also means you can't get a 20 and come back from death to consciousness. :)
Actually, you can. It says you take the better of your roll or 10. If you roll a 20, you take that.

Saph
2009-01-26, 06:34 PM
Actually, you can. It says you take the better of your roll or 10. If you roll a 20, you take that.

Huh. I could have sworn that that ability used to say "you can take 10 on death saves". Must have misread it.

Oh well, this is still useful for those who don't want to play a Warforged.

- Saph

Yakk
2009-01-26, 07:47 PM
With 2 rolls, taking the best...

D% chance of a single roll being a death token.
Z% chance of getting up
1-(D+Z) chance of staying down.

With 2 rolls:
D^2 chance of getting a death token.
2Z - Z^2 chance of getting up.

So let K be the chance of not just "skipping" = D^2 + 2Z - Z^2.

Let B = 1 mean no bonus to saves (makes the math easier), and each point past increases your bonus to saves.

Now, D = .5-.05B
Z = .05B

K = (.5-.05B)^2 + .1B - B^2/40
= 1/4 -B/20 + B^2/400 + B/10 - B^2/400
= 1/4 +B/20

Define X:= D/K. Then we get:
(.5-.05B)^2 / (1/4 + B/20)
(1/4 - B/20 - B^2/400) / (1/4 + B/20)

Now chart this from B = 1 (no bonus) to B = 10 (+9 bonus, auto-win) in excel, and you have the impact of the double-save belt.

Saph
2009-01-27, 08:33 AM
For those of us without Excel, what would that be? :P

- Saph

Yakk
2009-01-27, 02:10 PM
http://openoffice.org

Tygell
2009-01-27, 03:43 PM
Since you guys seem to be good at statistics, think you could help me figure a problem out? My next DM uses a weird stat generation, and I'm curious what the average scores would be.

Basically, we get one free 18, then roll 20d6, dropping the lowest 5. I can break that down to effectively roll 4d6 5 times, dropping the lowest roll each time, right?

Anyway, I just want to know what the average total will be of 20d6 dropping lowest 5. Thanks!

Kurald Galain
2009-01-27, 04:08 PM
Basically, we get one free 18, then roll 20d6, dropping the lowest 5. I can break that down to effectively roll 4d6 5 times, dropping the lowest roll each time, right?
No, it's better than that.

Anyway, I just want to know what the average total will be of 20d6 dropping lowest 5. Thanks!
Approximately 63.

20 dice net you an average of 70 pips. On average that'll include three ones and three twos, so you subtract 7.

Yakk
2009-01-27, 04:11 PM
No, that is better than 4d6 5 times, dropping lowest each time.

Because if you rolled 1166 on roll 1, then 6666 on roll 2... the 2nd 1 in the first roll would be 'stuck', and you'd be forced to lose the 6 from roll 2.

The easiest way to find the average of NdX drop K is to run a quick simulation and output the average. It should converge pretty quickly.

Reluctance
2009-01-27, 04:28 PM
See, that's what I thought, for exactly that reason. But people kept saying "3 in a row"...

"Three failures before you rest" takes into account being healed after having failed a death saving throw. If you fail a death save, get healed, and get knocked back down again, you have that much less of a buffer left.

Saph's question was "if left for dead, what are your chances of stabilizing on your own?" Since nobody's going to come back to clobber you again if you do get up, the carryover failures are a nonissue.

Artanis
2009-01-27, 04:55 PM
I know. I know what it says in the PHB, and I know about the carryover thing. What I meant was that everybody saying "three in a row" made me think that I had missed some vitally important errata or something that changed it to requiring three bad throws with no 10+ rolls in between them. I now know that that is not the case, and the rule is still as I thought.

Saph
2009-01-27, 05:04 PM
Define X:= D/K. Then we get:
(.5-.05B)^2 / (1/4 + B/20)
(1/4 - B/20 - B^2/400) / (1/4 + B/20)

Now chart this from B = 1 (no bonus) to B = 10 (+9 bonus, auto-win) in excel, and you have the impact of the double-save belt.

Hmm. Using the (1/4 - B/20 - B^2/400) / (1/4 + B/20) formula, I get results of:

+0 to saves: 0.66
+1 to saves: 0.40
+2 to saves: 0.19

Intuitively, that doesn't look right. +2 to saves and 2 rolls should give a far lower death chance than that.

Working it out with pen and paper, the result I got for the chance of death with a +2 to saves and 2 rolls was:

((Chance of one failed save squared) x 2.5) to the power 3. For a +2, this gives a result of 0.0287, or about 3%, which looks about right. Was I misreading your formula?

- Saph

Tygell
2009-01-27, 05:39 PM
Thanks for the help Kurald and Yakk! Sorry to have slightly hijacked your thread Saph.

On topic, I hadn't realized the death failures accumulated between extended rests. I'm pretty sure a friend at our game would be dead right now if we had known the correct ruling. Can I rule him as some sort of undead monstrosity now?

NecroRebel
2009-01-27, 06:10 PM
Thanks for the help Kurald and Yakk! Sorry to have slightly hijacked your thread Saph.

On topic, I hadn't realized the death failures accumulated between extended rests. I'm pretty sure a friend at our game would be dead right now if we had known the correct ruling. Can I rule him as some sort of undead monstrosity now?

They don't accumulate between extended rests, they accumulate between rests. Big difference. Resting 5 minutes resets the counter as surely as resting 6 hours. If you recharge your encounter powers, you wipe clean the death saves, so the fact that it only resets on rests is practically only applicable if you're getting dropped to below 0 multiple times in one battle.

Kurald Galain
2009-01-27, 06:10 PM
On topic, I hadn't realized the death failures accumulated between extended rests. I'm pretty sure a friend at our game would be dead right now if we had known the correct ruling. Can I rule him as some sort of undead monstrosity now?

Well, sure you can (assuming you're the DM there). Question is if you should, given that it is ultimately your (the DM's) mistake. In my opinion, certainly not - although you could make it a quest hook that some nether spirits are now chasing him.

Yakk
2009-01-27, 10:55 PM
With 2 rolls, taking the best, and a +X bonus to your saves, the chance you die before you get up from a healing surge is:
+0 0.307546875
+1 0.095533528
+2 0.0287229
+3 0.008
+4 0.001953125
+5 0.000384673
+6 5.27344E-05
+7 3.64133E-06
+8 4.55539E-08
+9 0

The natural log is:
-1.179127764
-2.348278018
-3.550060551
-4.828313737
-6.238324625
-7.863116472
-9.850243038
-12.52316181
-16.90436881
-infinity

which shows that your death chance shrinks faster-than-exponentially!