Highest Possible Crit Range?

[term1nally s1ck]

As long as there is a chance for failure, failure will occur within infinite bounds.

Prove it. If you provide me with a proof that it is guaranteed that the sequence will fail, I'll aplogise and stop arguing against it. I've proved using the mathematical principles defined to deal with infinities that this build makes an infinite number of attcks. Either find a flaw the the maths, or provide a proof that the opposite is true, don't just rely on shouting louder. You've not backed up your claims that I'm using infinity wrong, and you're not even trying to use any of infinity's properties, bar the one you think is relevant, despite multiple explanations as to why it really isn't.

[Gpope]

Since it's no zero, it happens.

Untrue. There is not actually any guarantee that everything that can happen will happen as you approach infinity. Because we can't actually predict what is going to happen in an infinite series, the only way to meaningfully discuss probability over infinity is in terms of limits.

Take for example a series of coin flips, straight 50/50 probability, and see how many times in a row it comes up heads. There is never at any point any guarantee that the streak will stop, no guarantee that the coin will ever come up tails. There is always a nonzero probability that it will come up heads one more time. It is impossible for a number X to exist such that the probability of getting X many heads in a row is 0, yet because the limit of the probability rapidly approaches 0 as X increases we can confidently say that the probability of getting infinitely many heads in a row is 0 even though it is always possible to continue flipping heads.

The limit of the possibility of getting infinitely many heads is 0; there is 0% chance of reaching infinitely many heads, even though the chance of the streak continuing is always higher than 0. The limit of the possibility of getting infinitely many attacks from the proposed attack loop (at least with the 6-20 threat and confirm) is 5/9; there is a 55.55...% chance of reaching infinitely many attacks.

As long as there is a chance for failure, failure will occur within infinite bounds. I don't need a bunch of math to show that as FACT.

Except you can't show it as fact at all. It might seem intuitively obvious, but you can't prove that failure will ever occur.

[Heliomance]

Just because we aren't throwing a bunch of equal signs around and using big hard-to-understand words, that doesn't mean our argument isn't valid. To assume so is a logical fallacy on your part.

I've said it before and I'll say it again. YOU ARE USING INFINITY WRONG.

You want a big number? Great. You have a 5/9 chance of getting a big number. Fantastic. You've proved that.

You want the attacks to go on ENDLESSLY, as in, never ending? Not possible with this build. PERIOD. As long as there is a chance for failure, failure will occur within infinite bounds. I don't need a bunch of math to show that as FACT.

Now, I suggest you either stop resorting to logical fallacy in order to win your argument, or admit defeat.

Personally, I would assume that the people throwing the symbols and "big hard-to-understand words" are more likely to be the people that know what the hell they're talking about. I realise this isn't always the case, and that a blanket assumption to that effect is an appeal to authority fallacy. I'm only in my first year of a maths degree, so I can't follow the maths required perfectly, but I'll do my best to translate.

The definition of a function f(x) (in this case the number of attacks generated) tending to infinity as a variable x (in this case the number of attacks made) tends to infinity is:
(∀ K>0) (∃ R>0) | (∀ x∈X) x>R ⇒ f(x)>K

Translated, this means:
For any value (which we will call K) greater than 0, we can find another value (let's call it R) greater than 0, such that for any value that it is possible for x (the number of attacks rolled so far) to take, if x is greater than the value we chose, R, then f(x) (the number of attacks generated so far) will be greater than the value K, no matter how high K is. So the fact that we have already established that there is a 5/9 chance of generating any number of attacks, no matter how high, given an arbitrarily high number of rolled attacks - in this case represented by setting R arbitrarily high - means that there is a 5/9 chance that the number of attacks will meet this definition. And as this is the definition of tending to infinity - this is what tending to infinity actually means, in strict mathematical terms - that means that there is a 5/9 chance that the result will go to infinity.

[term1nally s1ck]

Please pretty please tell me how to make the logic symbols...I want my logic symbols..

[Heliomance]

Please pretty please tell me how to make the logic symbols...I want my logic symbols..

Copy-pasted from here.

[Kelb_Panthera]

Just out of curiosity, outside of DnD and your mathematics courses how often do you guys actually get to use this level of mathematics? I'm not trying to be derogatory. I'm actually interested in learning several of the higher maths but if I'm going to be paying a university and not getting much use out of this sort of thing, I'm going to pace myself rather than go sign-up for a course tomorrow

If you feel that answering this post could de-rail the thread feel free to PM me

[Gan The Grey]

Alright, this is turning into a contest of egos, and it is obvious that people's egos are getting hurt. So let's try to keep this civil.

I'm not going to get into math to argue my point any further, because, as I've said already, math is not the problem. There is nothing wrong with your math. Your equations are well-thought out and precise. I completely understand what it is you are trying to say, I just believe you are misapplying evidence, or rather, apply the wrong evidence to the argument. I'm not the only one that has said this.

All I have to do to prove that this sequence will fail is to posit a string of 1's large enough to end the sequence. That's it. Within infinite bounds, all things that are possible will eventually occur, and as failure is a possibility, it will eventually occur, and the number of attacks will thusly fail to achieve infinity. I don't need math to show this, and this cannot be ignored either.

The thing about science is, you can prove your theory correct a million times, but it only takes one failure to prove it wrong. Just because your math comes out the way you want it to doesn't make it right. I'm sure you can say this goes both ways, and you would be right.

Terminally sick, any attitude I appeared to have was not directed at you. It was directed towards Agentpaper, in response to attitude received from him. So, don't take my responses personally. I'm not trying to yell with caps. I'm trying to emphasize what I feel are the most important parts of what I'm saying.

Let me ask you this though. I have conceded to the intelligence of yours and others' math multiple times now. Do you believe there is any merit to what I and others supporting my view are saying? You accuse me of being selective in my argument, but is not the most important aspect of infinity the fact that it literally has no end?

And let me ask one more question before I forget. I'm not even sure what number we are using any more to threaten a critical, be it 9 or 6 or 3...but what are the chances of tending to infinity if our critical threat range was simply 20, changing nothing else? Wouldn't that also tend towards infinity, even if the possibility was smaller?

[sambo.]

Arcane Duelist's False Keen specifically does not stack with either Improved Critical or keen.

errr, not exactly.....

actually: exactly NOT.

False Keenness (Ex): To create the idea that she is more effective than she really is, the arcane duelist can make her chosen melee weapon keen once per day per arcane duelist level. To activate the keen effect, she must subtract the weapon's normal threat range for critical hits from her attack bonus. For example, if the arcane duelist wields a rapier, she must subtract 3 from her attack bonus, since the rapier has a threat range of 18-20, to make it keen. The keen effect lasts for 1 round per arcane duelist level, and it does not stack with the keen edge spell or other magical effects that make a weapon keen. The ability does stack with the increased threat range granted by the Improved Critical feat and by a keen magic weapon.

it's actually a little ambiguous, but i read that as specifically stacking with Improved Critical OR a Keen weapon, but NOT a weapon with a Keen Edge spell on it.

in any case, i read it as specifically stacking with Improved Critical.

[Fortuna]

Gan, the problem is that we have shown that we surpass the finite numbers. We are talking transfinites now, and the argument as to the difference between those and infinity is very complicated indeed. Also, I believe that it would not tend to infinity, although I will need to check that by plugging the numbers into the general form of the limit when someone provides it (or just by waiting for someone else to find it). It may well do so, however, just with an incredibly small chance.

[Gpope]

All I have to do to prove that this sequence will fail is to posit a string of 1's large enough to end the sequence.

This only proves that the sequence can end, not that it will end, because:

Within infinite bounds, all things that are possible will eventually occur,

is a false assumption. It seems like it's the kind of thing that would be true, but it's not quite correct.

Here's another way of conceptualizing the problem: the longer the sequence goes, the larger it gets and the more 1s it takes to end it. If it continues infinitely, then it will take infinitely many 1s in a row (more or less) to break the sequence. You are correct in asserting that there is always a possibility that you will in fact roll infinitely many 1s in a row--but there also exists a possibility that you will roll infinitely many non-1s in a row, or roll an infinitely long mixture that does not contain enough 1s (which is basically infinitely many 1s) to break the chain. And since rolling 1s happens a lot less often than rolling critical threats that spawn additional attacks, the odds of getting infinitely many attacks outstrip the odds of getting infinitely many 1s.

[Kallisti]

I've played a similar build before, so there's some things I'd like to clear up. For the record, I'm AFB at the moment, so if someone has RAW to prove me wrong, then I'd like to see it. I hate being wrong, if you see me wrong correct me, please.

1. Lightning Mace is not dependant on the attack hitting or confirming a crit, only on it threatening a crit. Which means it only needs to roll within the threat range. Even a Nat 1 can threaten a critical, although it can never confirm or even hit. I'm pretty sure about this, but not 100%.

2. I know that there's an Eberron special material called crysteel. I don't know if it would count as a steel and crystal weapon, but I thought I'd mention it.

3. I believe there's a Devoted Spirit stance in ToB that lets you take 11 or so on attack rolls. I know that either Krimm Blackleaf or the Demented One has a homebrew discipline focused on the Lawful alignment that has a similar stance, although homebrew isn't really relevant. Thought I'd mention it.

Someone with the books will need to check these for me.

[Gan The Grey]

@ Random

Hmmm. I guess the issue I'm having here is that I view any defined number as finite. If you can declare a point in a number line, that point becomes finite. If you can reach a number, it is not infinite. Kind of like the argument of knowing where an atom is and how fast it's going. You can't have it both ways. Either you are dealing with an arbitrarily high number, or you are dealing with infinity.

I think you and I are arguing two different things actually. Correct me if I wrong here, but I think you are arguing for the potential to reach any given number you supply, whereas I am simply arguing for achieving an endless quality.


From Gpope

Here's another way of conceptualizing the problem: the longer the sequence goes, the larger it gets and the more 1s it takes to end it. If it continues infinitely, then it will take infinitely many 1s in a row (more or less) to break the sequence. You are correct in asserting that there is always a possibility that you will in fact roll infinitely many 1s in a row--but there also exists a possibility that you will roll infinitely many non-1s in a row, or roll an infinitely long mixture that does not contain enough 1s (which is basically infinitely many 1s) to break the chain. And since rolling 1s happens a lot less often than rolling critical threats that spawn additional attacks, the odds of getting infinitely many attacks outstrip the odds of getting infinitely many 1s.

I completely understand what you are saying here. And, at first glance, it would seem to make perfect sense. The problem here lies with how you are trying to give possibility concreteness. Yes it is possible to have more attacks generated than ones generated, but the reverse is also true, even if the chance is small that it will do so. You are right that the longer it goes on, the harder it gets to break the chain. I keep being told that I have to prove that it will break, yet no one has proved that it won't. Everyone is relying on the possibility that it won't, but wants me to provide proof that it 100% will break. That's not really fair.

I think, here, we are going to have to agree to disagree. Correct me if I'm wrong, but you are saying that even given an endless amount of time, not all possible circumstances will be reached, whereas I'm saying that given an endless amount of time, all possible circumstances will occur. Your opinion holds much more water than mine even if you consider a time span equal to the life of the universe, but time is not an issue within our DnD example, as all attacks will somehow be resolved in a span of six seconds. Really, I think this comes down to a difference of opinion that can't really be solved at all.

[Gpope]

Everyone is relying on the possibility that it won't, but wants me to provide proof that it 100% will break.

Well, yes. You're saying that there is a 100% chance it will break as it continues towards infinity, so it is imminently fair to ask you to prove that there is a 100% chance it will break. Moreover, you're talking about reciprocal outcomes here. If there is not a 100% chance it will break as it continues infinitely, then there is a possibility that it will continue infinitely.

I think, here, we are going to have to agree to disagree. Correct me if I'm wrong, but you are saying that even given an endless amount of time, not all possible circumstances will be reached, whereas I'm saying that given an endless amount of time, all possible circumstances will occur.

You've summed up the difference succinctly. The idea that everything will eventually happen as you go through infinitely many trials seems obvious, but it's misleading. The problem is that "all possible circumstances" include contradictory outcomes. It's possible that you will roll infinitely many natural 1s in a row, and it's also possible that you will roll infinitely many natural 20s in a row. These two outcomes are mutually exclusive--if you're rolling infinite natural 1s, you never start rolling natural 20s, so you can't have both of them occur in the same infinite sequence. Of course, the odds of either one happening are infinitesimally small, so we can safely rule them out even though there is no logical reason that either one couldn't happen.

You could, on the other hand, say with confidence that everything with a finite probability will eventually happen given infinitely many trials. You can rest assured that you will eventually roll a natural 1 at some point in infinitely many rolls, 100% of the time. You can even say with confidence that you are guaranteed to eventually roll any arbitrarily large number of natural 1s in a row that you care to name. But you cannot assume that you will roll infinitely many natural 1s in a row, and it takes infinitely many natural 1s to guarantee that the attack chain breaks.

[CockroachTeaParty]

I haven't bothered to read the whole thread, but I got the gist of it.

My question is, if this infinite critical hit attack thing worked, what would it look like?

Would it just be the character charging, his arms blurring into a spinning whirlwind of death? Would the blades be moving faster than the eye could see? Would the air combust?

[Gan The Grey]

Well, yes. You're saying that there is a 100% chance it will break as it continues towards infinity, so it is imminently fair to ask you to prove that there is a 100% chance it will break. Moreover, you're talking about reciprocal outcomes here. If there is not a 100% chance it will break as it continues infinitely, then there is a possibility that it will continue infinitely.

That's a good point, and it really just illustrates the differences in our opinions on possibilities within infinite bounds.

You've summed up the difference succinctly. The idea that everything will eventually happen as you go through infinitely many trials seems obvious, but it's misleading. The problem is that "all possible circumstances" include contradictory outcomes. It's possible that you will roll infinitely many natural 1s in a row, and it's also possible that you will roll infinitely many natural 20s in a row. These two outcomes are mutually exclusive--if you're rolling infinite natural 1s, you never start rolling natural 20s, so you can't have both of them occur in the same infinite sequence. Of course, the odds of either one happening are infinitesimally small, so we can safely rule them out even though there is no logical reason that either one couldn't happen.

You could, on the other hand, say with confidence that everything with a finite probability will eventually happen given infinitely many trials. You can rest assured that you will eventually roll a natural 1 at some point in infinitely many rolls, 100% of the time. You can even say with confidence that you are guaranteed to eventually roll any arbitrarily large number of natural 1s in a row that you care to name. But you cannot assume that you will roll infinitely many natural 1s in a row, and it takes infinitely many natural 1s to guarantee that the attack chain breaks.

But were ARE talking about finite probabilities, because at no point in the line does it require an infinite amount of 1's to break the chain. At any point, the number of 1's needed to break the chain is a specific number. So we can't discount this.

[term1nally s1ck]

I've played a similar build before, so there's some things I'd like to clear up. For the record, I'm AFB at the moment, so if someone has RAW to prove me wrong, then I'd like to see it. I hate being wrong, if you see me wrong correct me, please.

1. Lightning Mace is not dependant on the attack hitting or confirming a crit, only on it threatening a crit. Which means it only needs to roll within the threat range. Even a Nat 1 can threaten a critical, although it can never confirm or even hit. I'm pretty sure about this, but not 100%.

2. I know that there's an Eberron special material called crysteel. I don't know if it would count as a steel and crystal weapon, but I thought I'd mention it.

3. I believe there's a Devoted Spirit stance in ToB that lets you take 11 or so on attack rolls. I know that either Krimm Blackleaf or the Demented One has a homebrew discipline focused on the Lawful alignment that has a similar stance, although homebrew isn't really relevant. Thought I'd mention it.

Someone with the books will need to check these for me.

Aura of perfect order.

We have a winner. lets see, drop the weaponmaster, shove in swordsage 6, stick with a 9-20 crit threat, and take 11s. All you need to do now is buff your to-hit, which is less difficult. Just wow.

I take it this build would be accepted by all as an infinite loop, given that you hit on an 11 or less?


@Gan: Yes, there is some merit in having intuitive ideas about infinity, especially since you can get completely opposite ideas back, even with rigorous proofs.

http://en.wikipedia.org/wiki/Random_walk

Ironically, we're both right. There is simultaneously a <1 chance that the sequence terminates at a finite number, and a guarantee that the sequence terminates. I'm confused as hell how this is possible, but infinity is by its nature, counterintuitive.

Apologies to all for dsimissing so many people's ideas, apparently two completely opposite views can be correct in this case.

[term1nally s1ck]

Just out of curiosity, outside of DnD and your mathematics courses how often do you guys actually get to use this level of mathematics? I'm not trying to be derogatory. I'm actually interested in learning several of the higher maths but if I'm going to be paying a university and not getting much use out of this sort of thing, I'm going to pace myself rather than go sign-up for a course tomorrow

If you feel that answering this post could de-rail the thread feel free to PM me

Suprisingly, to me, the more I learned about maths, the more I was able to apply it to everyday stuff. There are lots of little questions people ask about hings, usually not expecting an answer to be possible, and once you've done some advanced maths, you know how the probvlem could be solved.

[Gan The Grey]

Aura of perfect order.

We have a winner. lets see, drop the weaponmaster, shove in swordsage 6, stick with a 9-20 crit threat, and take 11s. All you need to do now is buff your to-hit, which is less difficult. Just wow.

I take it this build would be accepted by all as an infinite loop, given that you hit on an 11 or less?


@Gan: Yes, there is some merit in having intuitive ideas about infinity, especially since you can get completely opposite ideas back, even with rigorous proofs.

http://en.wikipedia.org/wiki/Random_walk

Ironically, we're both right. There is simultaneously a <1 chance that the sequence terminates at a finite number, and a guarantee that the sequence terminates. I'm confused as hell how this is possible, but infinity is by its nature, counterintuitive.

Apologies to all for dsimissing so many people's ideas, apparently two completely opposite views can be correct in this case.

Nice research sir. +2 points. Now I'm going to take a look at that some more and wait for my brain to melt.

Oh...and +10 points to Kallisti for being a pimp. /Thread lol

[Gpope]

But were ARE talking about finite probabilities, because at no point in the line does it require an infinite amount of 1's to break the chain. At any point, the number of 1's needed to break the chain is a specific number. So we can't discount this.

But by the same token, at any point in the chain there is a finite, nonzero (nowhere near zero, once you get going) probability of the chain continuing.

Suppose we look at someone who just takes Lightning Mace and absolutely no other threat range modifiers or critical hit triggers. If they roll a natural 20, they keep rolling; if they ever roll anything than a natural 20, they stop. At any given point there's a 5% chance that you keep rolling, so there is no finite limit to the number of attacks that you can get from Lightning Mace. Yet I think we both readily agree that you're never going to get an infinite number of attacks from Lightning Mace, because the probability of getting an arbitrarily high number of natural 20s is infinitesimally small.

Similarly, with an attack loop that only stops on a natural 1 and expands itself otherwise, once you get enough attacks going you require an arbitrarily high number of natural 1s (not technically an infinite number of 1s, you are correct there) to stop, a probability that is also infinitesimally small as you go higher.

[Gan The Grey]

But by the same token, at any point in the chain there is a finite, nonzero (nowhere near zero, once you get going) probability of the chain continuing.

Suppose we look at someone who just takes Lightning Mace and absolutely no other threat range modifiers or critical hit triggers. If they roll a natural 20, they keep rolling; if they ever roll anything than a natural 20, they stop. At any given point there's a 5% chance that you keep rolling, so there is no finite limit to the number of attacks that you can get from Lightning Mace. Yet I think we both readily agree that you're never going to get an infinite number of attacks from Lightning Mace, because the probability of getting an arbitrarily high number of natural 20s is infinitesimally small.

Similarly, with an attack loop that only stops on a natural 1 and expands itself otherwise, once you get enough attacks going you require an arbitrarily high number of natural 1s (not technically an infinite number of 1s, you are correct there) to stop, a probability that is also infinitesimally small as you go higher.

Alright! Now we're getting somewhere!

This traditional use of Lightning Maces is really not much different than the build we are discussing. All you are doing is increasing the chances of additional attacks. Using this new example and your previous logic, you could say that you will infinitely NOT get an additional attack, simply because the chance is small. That's basically your argument for infinite attacks, right? You are going to get infinite attacks simply because chance favors you? The only REAL difference between the two is that chance favors you more and more the further in you go.

Let me ask this. Let's say the first time you try to set up your infinite attack routine, you roll a 1 each time you roll, until you run out of attacks. Doesn't that single circumstance right there disprove the idea that this build will afford you infinite attacks?

Oh, and as final proof for the validity of my argument, I cite 'Murphy's Law'.

'Anything that can happen, will happen, and at the worst possible time.'

So there.

[Fortuna]

No. It disproves that it will always afford infinite attacks.

[Gpope]

Alright! Now we're getting somewhere!

This traditional use of Lightning Maces is really not much different than the build we are discussing. All you are doing is increasing the chances of additional attacks. Using this new example and your previous logic, you could say that you will infinitely NOT get an additional attack, simply because the chance is small. That's basically your argument for infinite attacks, right? You are going to get infinite attacks simply because chance favors you? The only REAL difference between the two is that chance favors you more and more the further in you go.

That's essentially it. The key difference isn't the fact that you're increasing the threat range, though; it's the fact that you're abusing the rules to fit in two extra attacks for every critical threat. It doesn't matter whether your chance to keep rolling is 5% or 95%, with a fixed probability you're going to hit it sooner or later. But because you're constantly spawning off extra attacks, your chance to keep rolling continually goes up infinitely approaching a limit of 100%. If the limit was anything less than 100%, then you'd never be able to go infinite--like vanilla Lightning Mace, it would still be theoretically possible to keep going infinitely, but the odds would be infinitely small.

Also, Murphy's Law does not preclude the possibility that getting infinite attacks will end up being precisely the worst possible thing that could happen, and therefore must happen.

[Gan The Grey]

That's essentially it. The key difference isn't the fact that you're increasing the threat range, though; it's the fact that you're abusing the rules to fit in two extra attacks for every critical threat. It doesn't matter whether your chance to keep rolling is 5% or 95%, with a fixed probability you're going to hit it sooner or later. But because you're constantly spawning off extra attacks, your chance to keep rolling continually goes up infinitely approaching a limit of 100%. If the limit was anything less than 100%, then you'd never be able to go infinite--like vanilla Lightning Mace, it would still be theoretically possible to keep going infinitely, but the odds would be infinitely small.

Also, Murphy's Law does not preclude the possibility that getting infinite attacks will end up being precisely the worst possible thing that could happen, and therefore must happen.

But the limit never hits 100%, only APPROACHES it in smaller and smaller increments. It is always less that 100%. So then...not infinite. Correct? I might be misunderstanding this. If I am, how are you setting the limit at 100%? Technically, isn't the limit 95% (representing a 1 in 20) that increases incrementally with each successive attack?

Oh, that depends on your point of view. Yes, bad for the DM, but good for you since you intended for that to happen right? And since there is no DM in our example...then the worst possible thing to happen is for your build to not work as intended.

[Gpope]

But the limit never hits 100%, only APPROACHES it in smaller and smaller increments. It is always less that 100%. So then...not infinite. Correct?

Essentially, yes. The catch is, though, that limits work both ways. That's what I've been getting at with the plain Lightning Mace example. The probability of rolling X many natural 20s in a row approaches a limit of 0% very rapidly as X approaches infinity--but it never actually reaches 0, so Lightning Mace always has a nonzero possibility of generating an infinite number of attacks. Just as the self-reinforcing attack loop always has a nonzero possibility of terminating after a finite number of attacks (if you have a small number of attacks to begin with, there is a very noticeable chance that it will terminate simply by starting with multiple 1s--but once you reach enough attacks, the probability gets very, very close to 0 and only gets closer). In both cases, though, as you continue the sequence towards infinity it is for all practical purposes impossible to distinguish that nonzero probability from one that is actually zero. That's why it's safe to say that Lightning Mace cannot create an infinite attack loop on its own, even though there is no finite maximum number of attacks it can generate.

[Gan The Grey]

Essentially, yes. The catch is, though, that limits work both ways. That's what I've been getting at with the plain Lightning Mace example. The probability of rolling X many natural 20s in a row approaches a limit of 0% very rapidly as X approaches infinity--but it never actually reaches 0, so Lightning Mace always has a nonzero possibility of generating an infinite number of attacks. Just as the self-reinforcing attack loop always has a nonzero possibility of terminating after a finite number of attacks (if you have a small number of attacks to begin with, there is a very noticeable chance that it will terminate simply by starting with multiple 1s--but once you reach enough attacks, the probability gets very, very close to 0 and only gets closer). In both cases, though, as you continue the sequence towards infinity it is for all practical purposes impossible to distinguish that nonzero probability from one that is actually zero. That's why it's safe to say that Lightning Mace cannot create an infinite attack loop on its own, even though there is no finite maximum number of attacks it can generate.

Bah. I don't think you can just disregard something that is obviously there, regardless of how small the chance is. That seems like bad math to me, sir. Otherwise, no one would ever win the lottery, and life would have never evolved.

[Fortuna]

Very well then. We shall stop teaching calculus, resort to numerical approximations of all rates of change, and make a large amount of applied science a great deal more difficult. You do that in all of these things, so it obviously works somehow.

[Gpope]

Bah. I don't think you can just disregard something that is obviously there, regardless of how small the chance is. That seems like bad math to me, sir. Otherwise, no one would ever win the lottery, and life would have never evolved.

If you take an unlimited number of d20 rolls, there is a nonzero possibility of rolling an infinite number of any value from 1 to 20. It is possible you will roll nothing but 1s. It is also possible you will never roll a single 1, ever, even if you roll infinitely many times. In both cases, we can disregard these outcomes because their probabilities are infinitesimal--the chances are not merely "very small", they are so small that it is literally impossible to calculate how small they are other than to say that they are effectively 0.

[Gan The Grey]

Very well then. We shall stop teaching calculus, resort to numerical approximations of all rates of change, and make a large amount of applied science a great deal more difficult. You do that in all of these things, so it obviously works somehow.

Yes, it works so very well in fact that we've unlocked the secrets of the universe already. Oh, wait.

So what? Disregarding small numbers might work in some cases, but that doesn't mean it works here. BUT. Maybe I'm just inclined to strongly disagree with someone who doesn't possess any tact. Who knows?

Anyway, thanks for arguing with me Gpope. You present a strong case, and I totally get your point of view. I just...can't see it that way. I can't disregard those chances, however small. It just feels wrong to me, like eating a peanutbutter and mayonnaise sandwich. Just...ewwww.

/end me

[Gpope]

Anyway, thanks for arguing with me Gpope. You present a strong case, and I totally get your point of view. I just...can't see it that way. I can't disregard those chances, however small. It just feels wrong to me, like eating a peanutbutter and mayonnaise sandwich. Just...ewwww.

Saying that infinitesimal values can't be taken to equal 0 is one thing, and it does make intuitive sense. The problem is that the ramifications of doing so are self-contradictory. There is a nonzero chance of the chain terminating and a nonzero chance of the chain continuing. You're saying that the chance of the chain continuing can be disregarded (even though it gets arbitrarily close to 100% as you get high enough along the chain), yet the chance that the chain terminates cannot be disregarded (even though it gets arbitrarily close to 0%).

And like terminally sick points out, technically it's true--both possibilities have to be accounted for. The chain is simultaneously guaranteed to end and guaranteed to not end. The only true logical conclusion that can be drawn is that talking about probabilities in an infinite sequence gets silly awfully fast. But mathematically we can take illogical concepts like working with values of infinity and manipulate them to get useful, logical results.

[Heliomance]

I'm not going to get into math to argue my point any further, because, as I've said already, math is not the problem. There is nothing wrong with your math. Your equations are well-thought out and precise. I completely understand what it is you are trying to say, I just believe you are misapplying evidence, or rather, apply the wrong evidence to the argument. I'm not the only one that has said this.

All I have to do to prove that this sequence will fail is to posit a string of 1's large enough to end the sequence. That's it. Within infinite bounds, all things that are possible will eventually occur, and as failure is a possibility, it will eventually occur, and the number of attacks will thusly fail to achieve infinity. I don't need math to show this, and this cannot be ignored either.

The thing about science is, you can prove your theory correct a million times, but it only takes one failure to prove it wrong. Just because your math comes out the way you want it to doesn't make it right. I'm sure you can say this goes both ways, and you would be right.

Actually, maths is not science. In science, it's impossible to prove a theory true. You can merely show by experiment that it appears to be the case, and that nothing has disproved it yet. In maths, you can prove a result. You can prove it rigourously and beyond all doubt. The fact that the maths comes out right does indeed make it right.

I think you and I are arguing two different things actually. Correct me if I wrong here, but I think you are arguing for the potential to reach any given number you supply, whereas I am simply arguing for achieving an endless quality.

As I pointed out earlier, being able to reach any given number you supply is the exact mathematical definition of going to infinity. Infinity is pretty much defined as "Give me any number you like, no matter how big. Infinity is just a little bit bigger."

I think, here, we are going to have to agree to disagree. Correct me if I'm wrong, but you are saying that even given an endless amount of time, not all possible circumstances will be reached, whereas I'm saying that given an endless amount of time, all possible circumstances will occur. Your opinion holds much more water than mine even if you consider a time span equal to the life of the universe, but time is not an issue within our DnD example, as all attacks will somehow be resolved in a span of six seconds. Really, I think this comes down to a difference of opinion that can't really be solved at all.

No, we're saying that given an endless amount of time, all possible circumstances will be reached, but that that might not be enough to end the chain. Let me put it this way - assuming the chain goes on long enough, at some point I will roll 50 squigajillion natural 1s in a row. Notcontesting that. What we're saying is that by the time you roll 50 squigajillion natural ones in a row, there's a pretty good chance that you'll need 50 squigajillion and one, or more. What we're saying is not that given enough attacks, you won't roll an arbitrarily high number of natural 1s. What we're saying is that there'sa 5/9 chance that none of those arbitrarily long strings will be enough to end the chain at the point at which they occur.

The odds of your first 50 squigajillion rolls being natural 1s are incredibly small. The odds of you rolling 50 squigajillion natural 1s by the time you've generated 50 squigajillion spare attacks are also pretty small. Pat a certain point, it's vastly more likely than not that you will have more attacks than the number of natural 1s you get in a row. For this reason, although the cumulative probability of the attack terminating as you keep rolling never reaches 0, it also never reaches 1. It's a sum to infinity with a finite limit, just like the sum 1+1/2+1/4+1/8+...

As the number of attacks you make tends to infinity, the probability of any given attack ending the sequence tends to zero. As the number of attacks tends to infinity, the probability of any given attack or any of the attacks before it ending the sequence tends to 5/9.