# Thread: Highest Possible Crit Range?

1. ## Re: Highest Possible Crit Range?

Originally Posted by Heliomance
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.
Minor Nitpick: This is technically wrong as a premise. 'The fact that the maths comes out right' proves that your mathematical model is right. That doesn't necessarily prove your case right if your model does not adequately match the problem.

But eh.

2. ## Re: Highest Possible Crit Range?

I feel as though I've aided in making a monster.

His name will be Chav'Xaa of the Lost Heresy and he will gleam like a wildfire. Behold my horrific creation!

He's also the sun.

3. ## Re: Highest Possible Crit Range?

Originally Posted by Signmaker
Minor Nitpick: This is technically wrong as a premise. 'The fact that the maths comes out right' proves that your mathematical model is right. That doesn't necessarily prove your case right if your model does not adequately match the problem.

But eh.
However, in this case the problem is simple enough (simple being a relative term - the maths is horrible) that we can model it perfectly. The only factor that could possibly affect it that's not being taken into consideration is whether you're using a fair d20 or not, which I would think most people would agree is irrelevant to the theoretical exercise.

4. ## Re: Highest Possible Crit Range?

Originally Posted by lvl 1 sharnian
Also, if this works, can you threaten a critical hit on a natural 1?
Better Lucky than good... in the rogue's 3.5 splatbook

but It will eat a couple of feats in requisites... IMHO NOT worth it

Originally Posted by Signmaker
Minor Nitpick: This is technically wrong as a premise. 'The fact that the maths comes out right' proves that your mathematical model is right. That doesn't necessarily prove your case right if your model does not adequately match the problem.

But eh.
On a side note falsiability is not the only criteria for science, but I concur with the opinion that Math is not a science. Math is beyond science as much as philosophy and art are. In fact, math is a branch of Logics which in turn are a branch of philosophy.

Science is not the only valid way to represent and study the world. It's just the socially sanctioned and in pop culture the idealized way to do it.

5. ## Re: Highest Possible Crit Range?

Originally Posted by Gpope
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..
You don't seem to understand what infinity is. If you've flipped a coin an infinite amount of times, then it's come up heads an infinite amount of times, in a row, an infinite amount of times. It's also come up tails an infinite amount of times in a row, an infinite amount of times. Hell, it's even come up tails an infinite amount of times in a row, and then heads an infinite amount of times in a row, and then tails again and back and forth another infinite amount of times.

Once again: INFINITY IS NOT A LARGE NUMBER. You cannot apply normal math to it. Yes, you have a 5/9 chance of reaching any real number. For all intents and purposes, it may as well be an infinite loop. That doesn't mean it IS an infinite loop, though, and it never will be.

Anyways, I'm done here, unless you guys want to actually explain how something that has any chance of happening could possibly NOT happen after an infinite amount of time, then I'll listen, but if you're just going to keep throwing numbers and big, meaningless words at us and demanding we talk the way you do back, then that's fine too.

6. ## Re: Highest Possible Crit Range?

Once again, I ask you to look over the definition of tending to infinity I posted earlier. If there is a 5/9 chance of exceeding any real number, then there is a 5/9 chance of going to infinity, because that's what going to infinity means.

You're right in that you can't apply normal maths to infinity. What you can do, however, is apply the maths specifically designed to work with infinity to infinity. Which is what we have been doing.

7. ## Re: Highest Possible Crit Range?

Originally Posted by Heliomance
Once again, I ask you to look over the definition of tending to infinity I posted earlier. If there is a 5/9 chance of exceeding any real number, then there is a 5/9 chance of going to infinity, because that's what going to infinity means.

You're right in that you can't apply normal maths to infinity. What you can do, however, is apply the maths specifically designed to work with infinity to infinity. Which is what we have been doing.
it has a 5/9 chance of reaching any real number, not exceeding it. After that, it has some very low non-zero chance of reaching the next higher number, whatever that is.

8. ## Re: Highest Possible Crit Range?

Originally Posted by AgentPaper
You don't seem to understand what infinity is. If you've flipped a coin an infinite amount of times, then it's come up heads an infinite amount of times, in a row, an infinite amount of times. It's also come up tails an infinite amount of times in a row, an infinite amount of times. Hell, it's even come up tails an infinite amount of times in a row, and then heads an infinite amount of times in a row, and then tails again and back and forth another infinite amount of times.
Coming up heads an infinite number of times in a row precludes ever coming up tails. It can come up an arbitrarily large number of heads in a row--or it may never come up heads at all. It is theoretically possible to keep flipping that coin an infinite number of times and never come up heads. Of course, using that BAD UGLY MATH it is possible to show that this is effectively impossible even though there is a nonzero probability of it happening.

Originally Posted by AgentPaper
it has a 5/9 chance of reaching any real number, not exceeding it. After that, it has some very low non-zero chance of reaching the next higher number, whatever that is.
It follows that if there is at least a 5/9 chance of reaching any real number, then for any given number X there is also at least a 5/9 chance of reaching X+1. Therefore it has a 5/9 chance of exceeding any real number.

9. ## Re: Highest Possible Crit Range?

This is the argument I keep seeing: "infinite means everything will happen, so unless you have a 100% guarantee that you'll never miss, it won't go infinite."
I disagree.

Let me describe the build as clearly as I can:
It generates a certain amount of attacks.
If an attack crits, the build gets two more attacks.
If the attack misses, the build does not get an additional attack.

Nay-sayers will think to themselves: "I know what infinity is, eventually this build will miss enough times in the row and thus cannot ever achieve an infinite series of attacks." They will think this is the end of it.
They don't realize that the chance of total failure gets infinitely small, and they certainly don't bother to conceptualize the interaction between infinite time and infinitely small failure rates.

Here, let me prove with very basic mathematics and logic why the naysayers are wrong.

I have a magic bean. This bean has a very peculiar existence. In the first second of its existence, the bean has a 10% chance of ceasing to exist entirely. That is to say, after being created, only 90% of these beans survive for more than a second.
The second second of existence is even stranger! The beans that survive to their second second have about a 1% chance of ceasing to exist. That is to say, after being created, only about 89% of these beans survive for more than two seconds.
The third second of existence follows the pattern. That is to say, after being created, only about 88.9% of these beans survive for more than three seconds.
And so on, in that pattern.
Only about 88.88888889% of these beans survive for more than ten seconds.
Only 88.888[lot more 8s]889% of these beans survive for more than 10^100 seconds.
Does everyone see the pattern here?
No matter what time you posit, it's easy to see that over 88% of the beans make it to that time.

My question: As time goes to infinity, is it possible for any of these beans to still exist? In other words, is it possible for any of these beans to survive an infinite number of seconds?

Please note, at each second, there is some probability that a bean stops existing. At time 1, it was 10%, at time 2 it was about 1%, and it gets lower and lower (but never quite reaching 0).

Naysayers will immediately latch on and say "there's always a non-zero failure rate, therefore there's always a possibility that the bean will cease to exist, therefore no bean will ever exist for an infinite number of seconds."

Does that agree with rules governing the beans? The beans seem crystal clear, although the survival rate keeps getting lower over time, there's always more than an 88% chance that a bean survives to time N because of the interaction between the infinitely small failure rates and infinite time.

Edit: Thankfully, Kalirren below provided the math and a very nice explanation.

10. ## Re: Highest Possible Crit Range?

The only person in the past two pages who has known what (s)he has been talking about is Heliomance. (edit: and ocdscale, who ninja'ed me. gj.) Maybe we should try to get everyone on the same mathematical page here.

So what we really mean by "X will happen eventually" is that the the limit of the probability of X happening by time n as n goes to infinity goes to 1; this is equivalent to the probability of X not happening by time n going to zero.

So we can say that if you keep rolling a fair d20, you will eventually get a natural 1. The probability of never getting a natural 1 is by time n is 19/20^n. The limit of 19/20^n as n goes to infinity is 0.

In fact, you can ask, "Will a string of N natural 1s occur eventually?" Well, the chance of it occurring at any point is just 1/20^N. So the chance that a string of N natural 1s never occurs by time n is just (1-1/20^N)^(N-n+1), and the limit of this as n goes to infinity is 0. So the answer is, "yes, a string of natural 1s of arbitrary length will occur eventually."

The problem the intuitive arguments that have been presented by Gan the Grey, SensFan, and AgentPaper is the following, and I will set this aside in its own paragraph for emphasis:

The fact a string of natural 1s of arbitrary length will eventually occur somewhere in the sequence of rolls does not imply that the sequence of attacks is expected to terminate.

What do I mean by that? Consider now the following game:

Step 1: Roll a d20.
Step 2: If you get anything but a natural 1, go back to step 1.

Now you can ask, "when can we expect this process to terminate?" This question can be answered. If the probability that the process terminates on the n'th roll is denoted by P(n), then the expected value of the number of rolls before termination is

SUM_n [n * P(n)] = SUM_n [n*19^(n-1)/20^n]

because P(n) = (probability of not terminating on the n-1 steps before roll n) * (probability of terminating on roll n) = (19/20)^(n-1) * 1/20.

Now since the denominator grows faster than the numerator, that sum converges to some finite value. I'm too lazy to calculate the actual expected value, but trust me that it exists, and you can plug it into Mathematica or something and get a finite answer out of it if you want. I suspect that the answer is 20.

But, I find that I get a much better intuitive understanding of the situation by asking the question, "What's the expected number of additional rolls I gain for each roll I make?" This question is easy to answer: there are 20 possible outcomes, and on 19 of them I get to attack again, and on 1 of them I don't. So the expected number of additional attacks per attack is (19*1+1*0)/20 = 19/20. Note that this is less than 1. That is to say, for each attack we make, we don't expect to get that attack back. If we continue the process indefinitely, we'll eventually run out of attacks.

So now consider the process that we've been discussing:

Step 1: Roll an attack (and resolve its damage.)
Step 2: If you get a 6 or above on the attack roll, go to Step 1 -twice-.

Now you can ask the question "when can we expect this process to terminate?" As we've discussed before, if n is a natural number, adding up all of the probabilities that the process terminates on the n'th attack, over all n, gives you 4/9.

Thus the expected value of the number of attacks you make is

SUM_n [n * P(n)] = (something finite) * 4/9 + (something undefined in the real numbers) * 5/9 = undefined.

Thus the question "when can we expect this process to terminate?" -has- an answer: specifically, "never." I know this result is counterintuitive, but it's true. If that didn't help you understand, the next suggestion might.

We can, again, ask the question, "How many additional attacks do I get for each attack that I make?" Well, just like last time, there are 20 possible outcomes. On 15 of them (6 and above) I get 2 extra attacks. On the rest of them I whiff and get 0 extra attacks. The expected number of extra attacks I get per attack I make is therefore (15*2+5*0)/20 = 1.5. This is -greater- than 1. That is to say, for each attack I make, I can expect to get more than 1 attack back. Note that this already incorporates our failure chance.

It follows that as I keep attacking, I should expect to get more and more attacks, and/because these attacks accumulate faster than the misses do. Sure, I might get unlucky on my "first few tries" and lose out. That's what the 4/9 probability of terminating means. But that's not what one would expect would happen. One would -expect- that the number of attacks continues to grow without bound. And the more attacks I have under my belt, the less I am vulnerable to "a few" consecutive losses. (And by "a few" or "the first few" I really mean "any finite number you care to name.")

Note that all of this argument really has very little to do with whether or not a sequence of natural 1s of arbitrary length will crop up eventually. It will. No one is contesting that. What we are trying to let you understand is that it also doesn't matter. Yes, you can go ahead and compute the probability that you get a sequence of natural 1s of length N, and you can compute the probability that a sequence of natural 1s of length N terminates the attack sequence, and you can multiply those two probabilities together and add them over all N and take the limit as N goes to infinity to get the probability that the attack sequence terminates at all, and you will get 4/9. The probability that the attack sequence terminates does not go to 1: therefore, one cannot expect the attack sequence to eventually end.

11. ## Re: Highest Possible Crit Range?

The only problem is, I've already found a proof that BOTH arguments are true.

My proof, as stated above, that the probability of it terminating in finite time is 4/9.

And a proof which is described (but not rigorously stated, you can find that on there internet elsewere) here.

Basically, given a random walk, where you start at n, and at each step you have a probability p of subtracting 1, and a probability q (<1-p) of adding 1, and a probability 1-p-q of staying at the same number, the walk has probability 1 of reaching all points on the number line, including 0, where in this case, the process would end.

So simultaneously, there's a probability of 4/9 that the process ends in a finite time, and a probability 1 that it ends.

Infinity is WEIRD.

But anyway, for those who want to see the build, it's currently Fighter 2 Swordsage 6 DoD 8, has a crit range of 9-20, and can take 11 on attack rolls. This one is TRULY infinite.

12. ## Re: Highest Possible Crit Range?

Originally Posted by term1nally s1ck
The only problem is, I've already found a proof that BOTH arguments are true.

My proof, as stated above, that the probability of it terminating in finite time is 4/9.

And a proof which is described (but not rigorously stated, you can find that on there internet elsewere) here.

Basically, given a random walk, where you start at n, and at each step you have a probability p of subtracting 1, and a probability q (<1-p) of adding 1, and a probability 1-p-q of staying at the same number, the walk has probability 1 of reaching all points on the number line, including 0, where in this case, the process would end.
I'm pretty sure that your conclusion does not follow. A random walk with static probabilities is not analogous to the situation presented.

Edit: Removed probably erroneous reasoning to avoid confusion, but leaving conclusion. Will post below after some thought and looking over the proof.

Edit 2: Or the proof is not applicable to this situation.

13. ## Re: Highest Possible Crit Range?

Try again, the walk is based on the number of attacks, and each step is 1 attack.

I make 1 attack, I have a 3/10 chance of not critting (-1 attacks), a 133/200 chance of critting and confirming (+1 attack), and a 7/200 chance of critting but not confirming (0 change.)

Situation is identical, if you count by attack.

14. ## Re: Highest Possible Crit Range?

Originally Posted by ocdscale
Does that agree with rules governing the beans? The beans seem crystal clear, although the survival rate keeps getting lower over time, there's always more than an 88% chance that a bean survives to time N because of the interaction between the infinitely small failure rates and infinite time.
While there may always be more than an 88% chance that a bean survives to time N, that doesn't mean that a particular one will. That is, even if an attack chain can go on infinitely, that doesn't mean that it will this time.

15. ## Re: Highest Possible Crit Range?

Of course the obvious solution to the issue is that the attack cannot go on to infinity, because eventually the DM will get annoyed and kill you.

16. ## Re: Highest Possible Crit Range?

Originally Posted by term1nally s1ck
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.
As much as I'd like to accept this, I believe Aura of Perfect Order only allows 1 roll a round to be treated as a natural 11 so while it could start the chain, it wouldn't be able to continue it as the attacks continue within the same round.

Perhaps when this occurs, tiny shockwaves erupt all around the body, severing muscles and organs or something.

17. ## Re: Highest Possible Crit Range?

@Kalirren and odcscale

Yes, I understand the model you're pushing. Thank you for assuming I was ignorant of it after saying I had understood it, and claiming that everyone who doesn't agree with you doesn't know what they're talking about. I'll ignore that bit for your sake.

All of that math proves that you have X% chance of reaching any real number. (about 5/9 in this case) Infinity is not included in any real number. This is why the attacks can become arbitrarily high, to the point where counting them is impossible. Name any really high number you want, and there's a 5/9 chance of you getting that many attacks or more. The problem is, infinity is not a number.

If you've rolled a dice an infinite amount of times, that means you've rolled a 1 an infinite amount of times in a row, which means that the chain has been broken. Since the chain has been broken, that means it's ended, which means it can't be infinity, since infinity has no end by definition.

The same thing goes for your bean example. If you wait any amount of time, there will be about an 89% of the beans left at any given time. If you then wait an infinite amount of time, ALL of the beans will be gone, because at any step in time, each bean has a chance of ceasing to exist, and since that chance (how much the chance is is irrelevant) is happening an infinite amount of times, it has a 100% chance of happening due to the definition of infinity.

18. ## Re: Highest Possible Crit Range?

Originally Posted by AgentPaper
If you've rolled a dice an infinite amount of times, that means you've rolled a 1 an infinite amount of times in a row,
Except it doesn't. You've rolled 1 an arbitrarily high number of times in a row, but as has been repeatedly established by people on both sides of the argument, an arbitrarily high number is not the same thing as an infinite number. An infinitely long streak of 1s is theoretically possible, since any given roll has a fixed chance to continue coming up 1, but it is infinitely improbable (whereas a streak of arbitrary length has a finite probability and in theory will eventually happen over an infinite span of time).

More importantly, an infinitely long streak of 1s precludes the possibility of an infinitely long streak of any other number; there's no reason that rolling an infinite streak of 1s should be any more likely than rolling an infinite streak of 2s, or 3s, or 20s, and yet you can have only one infinitely long streak (the first streak would have to have a finite end before a second streak could begin.)

19. ## Re: Highest Possible Crit Range?

Originally Posted by AgentPaper
All of that math proves that you have X% chance of reaching any real number. (about 5/9 in this case) Infinity is not included in any real number. This is why the attacks can become arbitrarily high, to the point where counting them is impossible. Name any really high number you want, and there's a 5/9 chance of you getting that many attacks or more.
For the Nth time, that is precisely how going to infinity is defined. I've already given the strict mathematical definition and translated it into English. If for any number no matter how high, we can get more attacks than that, then we can get infinite attacks by definition.
The same thing goes for your bean example. If you wait any amount of time, there will be about an 89% of the beans left at any given time. If you then wait an infinite amount of time, ALL of the beans will be gone, because at any step in time, each bean has a chance of ceasing to exist, and since that chance (how much the chance is is irrelevant) is happening an infinite amount of times, it has a 100% chance of happening due to the definition of infinity.
The definition of infinity doesn't actually contain anything about containing every possible result in it. It's entirely possible for a function to tend to a finite value as x tends to infinity, and there are plenty of examples. (e^x-e^(-x))/(e^x+e^(-x)), for example, tends to 1 as x tends to infinity. The example with the beans, if we take f(t) as the proportion of beans left after time t, tends to 8/9 as t tends to infinity.

20. ## Re: Highest Possible Crit Range?

Originally Posted by term1nally s1ck
Try again, the walk is based on the number of attacks, and each step is 1 attack.

I make 1 attack, I have a 3/10 chance of not critting (-1 attacks), a 133/200 chance of critting and confirming (+1 attack), and a 7/200 chance of critting but not confirming (0 change.)

Situation is identical, if you count by attack.
Well, but then the random walk is biased. I thought that any given state (including 0, or the state corresponding to any finite number for that matter) is only guaranteed to be recurrent if the random walk is unbiased, and is transient otherwise. So our first proof that the chain of attacks marches off to infinity would be correct...right?

21. ## Re: Highest Possible Crit Range?

Originally Posted by Gpope
there's no reason that rolling an infinite streak of 1s should be any more likely than rolling an infinite streak of 2s, or 3s, or 20s, and yet you can have only one infinitely long streak (the first streak would have to have a finite end before a second streak could begin.)
That statement clinches the argument for finite numbers, I would think. That is to say, there is a distinct probability of reaching any finite number of attacks with the given constraints.

I think the issue that some people have (I am quite easily one of them) is the knowledge that if any finite number of 'steps' has a distinct probability of stopping right then and there, that progression to infinity should be impossible, because eventually probability will not behave as the norm, using the a sort of twist on the Gambler's Fallacy to interpret that there will be periods in which a string of finite ones will terminate all of the remaining attacks despite 'expected' growth. After all, if we're talking about infinity, there are an infinite number of intervals in which this improbability can occur.

However, one can easily state that the step progression 'grows without bound' and 'tends to infinity'. The idea behind this is that while the above may occur more readily for low numbers, it gets progressively more difficult the more steps occur. So much so that the chance at extremely high numbers more or less 'vanishes'. This would be where series come in to play, and that is how the 5/9 number is achieved; most of that probability comes from relatively low finite numbers. After all, calculus teaches us at least that much. Amusingly, interpretation through series models the aforementioned Gambler's logic quite efficiently, despite protests.

But of course, actually interpreting whether or not there can exist an infinite number of attacks at a readily determinable probability is confusing, especially without the knowledge of upper-level math courses. It's especially difficult to teach the concept of series to those not mathematically versed (often still to those that are, such as myself).

Yes, there was no real point to my post. I just felt like summing up the current hurdles that needed to be jumped.

22. ## Re: Highest Possible Crit Range?

Originally Posted by Kalirren
Well, but then the random walk is biased. I thought that any given state (including 0, or the state corresponding to any finite number for that matter) is only guaranteed to be recurrent if the random walk is unbiased, and is transient otherwise. So our first proof that the chain of attacks marches off to infinity would be correct...right?

Actually, I've got my hands on a proof that it's guaranteed to terminate even in the biased case. Very very clever maths, and very complicated, but works. If you're on a uni connection, you may have free access, otherwise, it's pay to view.

23. ## Re: Highest Possible Crit Range?

Slide me the link then, I'll check it out next time I'm on campus.

It just seems strange that you would be forced to terminate, but you -could- think of it as Gambler's ruin by playing a game with finite expected value against a bank with an infinite bankroll. But if the probability is 1 that you reach 0 by step N as N goes to infinity, and you terminate when you get to 0, how can the sum of probabilities over all the natural numbers n that you make exactly n attacks sum to anything other than 1? We know it doesn't. Or at least you claim so: we'd need to look at your 5/9 proof. (Correction: I suppose we know it doesn't for the case where the entire pool of attacks doubles if -any- of the attacks hit and crit, in which the expected value of the game is not constant...the theoretical formation wasn't quite right...)

24. ## Re: Highest Possible Crit Range?

Originally Posted by term1nally s1ck
And yes, the build can go infinite, I spent a while writing out some of the maths to prove it. the 7-20 threat range which can hit on a 2 has a 60/133 chance to never run out of attacks, with each attack it makes. It doesn't hit with every attack, and it's not a guarantee that it'll go infinite....but if you get 3 attacks, and you hit with a 2 on all of them (Blood on the Water, hit rats. a lot.) then you get an 83.5% chance to go infinite that turn.
Once again, to reiterate what everyone else has said, this is not infinite. It's arbitrarily high. If there's ANY chance it can end, it cannot go infinite. There is not a magical limit that you can break on exponential numbers that cause infinity. If you're getting infinity, you are suffering a rounding error.

When discussing infinity, EVERY last digit becomes significant. This means that if a repeating number comes up, you'd have to map the decimal places out to infinity to do the math.

You can make it so each attack that successfully hits makes it more and more likely the chain will not end on the next iteration, but it cannot and will not go infinite. You cannot remove the inevitability that you will, given infinite time, eventually roll all nat 1's.

25. ## Re: Highest Possible Crit Range?

Time for some constructive input (for a change, as opposed to babbling).

Soon to be presented is more or less a tree diagram which should adequately model relative probabilities for finite numbers. In observing the rate of change between certain intervals, a convergent probability of failure at any stage should be attained (whether or not this works for infinity is not in my realm to know).

Stage #
1. A d20 is rolled.
2. For any value other than 1, perform these instructions twice.

In DnD terms, this would be equivalent to fumbling an attack on a 1, and gaining two new attacks plus resolving your current attack on any number other than 1.

Feel free to draw the following out.

Stage 1
There is a 1/20 chance of stagnation in respect to this stage and this stage only.
There is a 19/20 chance of two branches occurring.

So. We have a 5% (1/20) chance of the attack stream ending at stage 1.

Stage 2
There is a 1/400 chance of stagnation in respect to this stage and this stage only.
There is a 38/400 chance of two branches occurring.
There is a 361/400 chance of four branches occurring.
Therefore in respect to this stage only, there is a .25 percent chance of failure.

Now then, the probability of not making it to stage 3 (that is to say, to stop at either stage 1 or stage 2) is 1/20 + (19/20 x 1/400). This is equal to 419/8000, or 5.2375%. So we have a probability increase of .002375 from the previous stage.

Stage 3
Stage 3 Preface:
It gets a bit...expansive at this stage. Depending on the results of stage two, there can be either two or four rolls resolved in stage 3. For the sake of brevity, I'll only make note of the failure percentages.

In the occurrence of two rolls, there is a 1/400 chance of failure with respect to this stage only.
In the occurrence of four rolls, there is a 1/160000 chance of failure with respect to this stage only.
Therefore in respect to this stage only, there is a .0243% chance of failure.

The probability of failure before stage 4 is then:
(1/20) + (19/20 x 1/400) + ((38/400 x 1/400) + (361/400 x 1/160000)) = 3367561/64000000, or 5.2618%. This is a probability increase of approx. .000243 from the previous stage.

So what can be initially gleaned? That the chance of failure does increase with time, but really freaking slow. That is to say, it should converge before 1.

Let's try the following for the sake of simplicity. I flip a coin. Heads, nothing happens. Tails, I get two more flips.

Stage 1: 1/2 chance of total failure, 1/2 chance of two more flips.
Stage 2: 1/4 chance of stage failure, 1/4 chance of two more flips, 1/2 chance of four more flips. So we're at 75% failure by stage two (or before stage three, whichever mental reasoning you prefer)
Stage 3: Stage failure is now 1/16 (two-flip failure) + 1/32 (four-flip failure) = 3/32. So we're at 84.375%. Were we considering the geometric series (1/2+1/4+1/8.....=1), we would currently be at 87.5. We are not, which implies that this example converges before 1. It has quite a high chance of failing, but failure is not 'guaranteed'.

26. ## Re: Highest Possible Crit Range?

Originally Posted by Dairun Cates
Once again, to reiterate what everyone else has said, this is not infinite. It's arbitrarily high. If there's ANY chance it can end, it cannot go infinite. There is not a magical limit that you can break on exponential numbers that cause infinity. If you're getting infinity, you are suffering a rounding error.

When discussing infinity, EVERY last digit becomes significant. This means that if a repeating number comes up, you'd have to map the decimal places out to infinity to do the math.

You can make it so each attack that successfully hits makes it more and more likely the chain will not end on the next iteration, but it cannot and will not go infinite. You cannot remove the inevitability that you will, given infinite time, eventually roll all nat 1's.
If the chance the attack string ends on the first round is 1/4
And the chance the attack string ends on the second round is 1/8
And the chance the attack string ends on the third round is 1/16
And so on in that fashion.

Once started, what is the probability this attack string will end?
In other words, what is 1/4 + 1/8 + 1/16 + 1/32 and so on into infinity?
I say the sum of all the probabilities of failure, as the rounds go into infinity, is 50%. Therefore there is a 50% chance of failure, and a 50% chance the attack string will not fail.

You argue there is a 100% chance this attack string will fail eventually because every round has a non-zero chance it will fail.
I invite you to show me how to add 1/4 + 1/8 + 1/16 ... in order to sum it up to 1 (100%).

27. ## Re: Highest Possible Crit Range?

Another thing to address: infinity as a 'place', that constantly moves. Those who say "you will eventually roll enough ones" more or less use this as proof that the stream will end, because you cannot 'catch up' to infinity.

Back to the impish pixie riddle I threw forward so long ago. Two balls in, lowest ball out at each interval. At infinity you're left with none. However, were the riddle to be stated that the highest ball were removed, you'd have an infinite number of balls left at infinity, the odd real set of numbers. That is to say, though the two 'actions' seem the same (2 in, 1 out), their treatment at infinity is quite different, which goes to show the danger of treating infinity as a moving place that you analyze at. This is why you will hear 'tends to infinity' or 'grows without bound' rather than acknowledging 'hitting' infinity, because it's frankly not practical and sometimes logically contradictory.

28. ## Re: Highest Possible Crit Range?

Originally Posted by Dairun Cates
Once again, to reiterate what everyone else has said, this is not infinite. It's arbitrarily high. If there's ANY chance it can end, it cannot go infinite. There is not a magical limit that you can break on exponential numbers that cause infinity. If you're getting infinity, you are suffering a rounding error.
Once again, to reiterate what everyone who understands the maths has said, you're wrong. Infinity does not work how you think it works.

29. ## Re: Highest Possible Crit Range?

Originally Posted by ocdscale
You argue there is a 100% chance this attack string will fail eventually because every round has a non-zero chance it will fail.
I invite you to show me how to add 1/4 + 1/8 + 1/16 ... in order to sum it up to 1 (100%).
The chance of a coin flip being heads is 50%. The chance of a coin flip being heads a second time is 25% because 1/2 times 1/2 is 1/4th. This is basic math.

Let's put this in decimals though. .5 x .5 = .25

So, let's the growing problem here. The chance of SUCCESS on turn one and two are 3/4 and 7/8 respectfully.
So, the probability of both events occurring is...
3/4 x 7/8 = 21/32
or .75 x .875 = .65625
That's the total probability that both of those outcomes will occur.

So, we take 21/32 (the probability of the first two turns) and multiply it by the chance of success 15/16. So..
21/32 x 15/16 = 315/512
Or...
.65625 x .9375 = .615234375

Then...
315/512 x 31/32 = .59600830078125 or 9765/16384

Eventually, the numbers start getting cut off by your calculator as they get smaller and smaller, but to get a TRUE statistic, you MUST include every decimal.

So yes. The DIFFERENCE between each probability each turn is going to to be smaller and smaller because of the exponential effect, but at no point will the probability GROW, because you never have a 100% chance or more of success. That number keeps going down and down. There's ALWAYS a chance for success on the next go, but the chances of it to actually keep going in the long run gets lower and lower.

Of course, the simple proof that you will eventually roll all ones. An infinite loop assumes infinite rolls. In infinite rolls, you will exhaust EVERY last possibility and permutation of dice rolling possible. One of those permutations is all 1's. You cannot remove this possibility. So, in your "infinite" rolls the possibility will occur and end the loop.

Just because you're less likely each turn to fail that turn, the chance is always there, and probability will catch up to you. Infinity cannot be caught up to. It is not an arbitrarily high number. It is a concept. You cannot ensure an infinite loop unless there is no other option. You HAVE to reach 100% probability to reach infinity. The numbers you're showing will add up to 1 only when you have done an INFINITE number of iterations.

In other words, until you reach 1/infinity as the current probability of failure, your fractions DO NOT add up to 1. Since that takes infinite rolls, one of them will inevitably fail because you have an infinite number of rolls to get there.

30. ## Re: Highest Possible Crit Range?

For all those words, you didn't answer my question.

What is 1/4 + 1/8 +1/16 + 1/32 . . and so on, what is it equal to?

You don't even need to explain your answer or show your work, I just want to know what you think the answer is.

Edit: It occurs to me that maybe you did try to answer the question. If that is the case, then there is no need to reply to this post, your answer can stand for itself.

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