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Thread: Highest Possible Crit Range?
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2009-11-29, 02:30 AM (ISO 8601)
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Re: Highest Possible Crit Range?
would casting single crystal blades out of iron be a bit too cheesy? (in order to qualify for both weapon properties)
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2009-11-29, 02:40 AM (ISO 8601)
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Re: Highest Possible Crit Range?
The problem is that you need "math equation bombs" to talk about infinity in any kind of rigorous fashion.
1. The basic premise of infinity is something that never ends. If there is a problem with this definition, webster it.
2. Infinity is NOT a specific number. It is an idea, and sometimes a state.
3. In order to achieve the ability to continue on for an infinite amount of time, there must be a 100% chance or greater to do so.
4. Infinity either IS or IS NOT. There is no middle ground. Either you have a 100% chance of being infinite, or you are finite.
2. Fine
3. Wrong. By means of "math equation bombs", it has been demonstrated that the build has a chance to achieve a number of attacks which is not finite, that is, which is infinite. Specifically, it has been shown that as the number of attacks which has been made increases arbitrarily, the likelihood that the system has collapsed DOES NOT become arbitrarily close to zero. It instead becomes arbitrarily close to something very close to 5/9. Therefore, it is possible to plug in the highest natural number, that is aleph-0, as the number of attacks which has been made and recieve a probability that the system has not collapsed a non-zero number. Which means that there is a chance that the system will achieve an infinite amount of attacks, specifically about 5 times in nine starting attacks (an interesting side effect is that each attack which is spawned has the same chance to go infinite, but that is beside the point).
4. See 3.If I creep into your house in the dead of night and strangle you while you sleep, you probably messed up your grammar.
I'm always extremely careful to hedge myself against absolute statements.
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2009-11-29, 02:48 AM (ISO 8601)
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Re: Highest Possible Crit Range?
Wrong. There is no 'chance'. It either is, or is not. The failure chance gets smaller the further and further in you go, but that chance is ALWAYS there. There is no getting rid of it. And as long as it is there, the possibility of going on forever is 0.
You are treating infinity as a specific, though massively large, number. Something that infinity IS NOT. Infinity is not 100. Nor is it 10^10000^10000000. Nor is it aleph-0. It's not even a number. It's an idea.
I'm not arguing that you will get a lot of attacks. But alot does not equal infinite. Either their is no chance of failure, or there is.
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2009-11-29, 03:01 AM (ISO 8601)
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Re: Highest Possible Crit Range?
I think what you're saying is that, as the number you're trying to reach increases, the chance you will get to that number approaches 5/9. Right? Unfortunately, this has no correspondence to infinity, since infinity isn't actually a number.
5e Homebrew: Death Knight (Class), Kensai (Monk Subclass)Excellent avatar by Elder Tsofu.
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2009-11-29, 03:05 AM (ISO 8601)
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Re: Highest Possible Crit Range?
Okay the way I found to treat infinities in situations like this is to state the following:
Assertion 1: Infinity is not a number.
Assertion 2: D&D works with numbers.
Therefore: Nothing in D&D is truly infinite.
Therefore: Infinity will instead represent all permutations of chance simultaneously.
Therefore: If there is a permutation that would end the supposed chain, no matter how mathematically improbable, comparing it to infinity means that it will occur.
Therefore: An Infinite loop must show that all possible permutations of an event have no chance of not seeding the next permutation (i.e, you must prove that the build will continue under the most statistically unfavorable of conditions).
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2009-11-29, 03:08 AM (ISO 8601)
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2009-11-29, 03:09 AM (ISO 8601)
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Re: Highest Possible Crit Range?
It's possible that there's a point where it tends to be infinite.
If we let the number of attacks be N, then the number of 1s that must be rolled in a row to break the chain will be f(N). That is, if we roll 1 attack, it will take 1.1 1s in a row to break the chain (assuming the average attack nets us 1.1 attacks afterwords). The probability of rolling f(N) 1s in a row is easy, it's (1/20)^(f(N)) or 1/(20^f(N)).
Now we can examine end behavior simply:
if the limit as f(N) approaches infinity is infinity, then the loop is infinite in that as you get more and more attacks it will be less and less likely to break out of the loop. If you get 1.1 attacks back for every attack you get in, then f(N) is 1.1^N, and as N approaches infinity, 1.1^N approaches infinity, so the loop is infinite in the sense that its limit is infinite.
more importantly: because the probability of rolling what you need to roll decreases as N gets large, it's not as simple as "infinite monkeys" scenario, because the "infinite monkeys" are making the proverbial works of shakespear even less likely. The probability that ALL monkeys on typewriters type the works of shakespear decreases as you get more and more monkeys.
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2009-11-29, 03:15 AM (ISO 8601)
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Re: Highest Possible Crit Range?
The problem here is you are talking about certainties in a realm of possiblities. It is not certain that you will always get 1.1 attacks back. That's an average. Averages don't take into account the ten times out of twelve I rolled less than a five on a d20. If we went by that....
If winning the lottery was a 1 in a million chance, and I bought 1 million tickets, I would win. But it doesn't work like that. (Lets not get caught up on this example. I think you know what I mean.)
If you can tell me that there is NO chance of rolling more 1's than any of the other numbers combined, then you are right. Unfortunately, we both know that isn't true.Last edited by Gan The Grey; 2009-11-29 at 03:22 AM.
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2009-11-29, 03:39 AM (ISO 8601)
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Re: Highest Possible Crit Range?
As I see it, there are two arguments going on here. One is based on a conceptual argument, and one on a rigorous mathematical proof. I humbly request, sir, that you attempt to state your argument mathematically as I tried to (and failed), by directly countering any step in the proof that this does in fact produce an infinite number of attacks some of the time.
And incidentally, aleph-0 is not infinity. I beg your pardon. Aleph-0 is a transfinite number, and if you insist then I will go back and edit every single one of my posts to make it clear that I am talking only about a number which emulates infinity in EVERY WAY IMAGINABLE.Last edited by Fortuna; 2009-11-29 at 03:44 AM.
If I creep into your house in the dead of night and strangle you while you sleep, you probably messed up your grammar.
I'm always extremely careful to hedge myself against absolute statements.
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2009-11-29, 03:50 AM (ISO 8601)
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Re: Highest Possible Crit Range?
The problem is not with your math. I applaud your math, sir. Your math freakin rocks.
The problem is a problem of logic. Something cannot 'possibly' be infinite. It's a true or false, not a percentage.
Here's what little math I can muster to prove that infinity and percentages cannot function together:
100% of infinity = infinity
99% of infinity = infinity
infinity = infinity K that works.
99% = 100% that doesn't work.
Hell, 1% of infinity is still infinity, except that percentages are a measurement of numbers, and infinity is NOT a number, so frankly I can't even say that.
Either something goes on forever 100% of the time (100% = true) or 0% of the time (0% = false). Anything in between doesn't fit the boundaries of the equation. For a true/false, there is no in between.
And incidentally, aleph-0 is not infinity. I beg your pardon. Aleph-0 is a transfinite number, and if you insist then I will go back and edit every single one of my posts to make it clear that I am talking only about a number which emulates infinity in EVERY WAY IMAGINABLE.
To quote the Wikipedia:
The aleph numbers differ from the infinity (∞) commonly found in algebra and calculus. Alephs measure the sizes of sets; infinity, on the other hand, is commonly defined as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or an extreme point of the extended real number line.Last edited by Gan The Grey; 2009-11-29 at 04:12 AM.
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2009-11-29, 03:57 AM (ISO 8601)
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Re: Highest Possible Crit Range?
I just accepted that we could not work with the truly infinite. There, you win. Working with the truly infinite is, as far as I am aware, at the boundaries of modern mathematics. Therefore, let's see whether we can achieve Cantor's poor imitation thereof. You say that the problem is not with math, but with logic. Your argument in that post says nothing to me except that dividing by infinity gives nonsensical answers. Where did we divide by infinity? If we did not, then I do not think that your argument holds water. And also, all that your last statement says is that any average value that we extract will also be nonsensical.
Last edited by Fortuna; 2009-11-29 at 03:57 AM.
If I creep into your house in the dead of night and strangle you while you sleep, you probably messed up your grammar.
I'm always extremely careful to hedge myself against absolute statements.
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2009-11-29, 04:10 AM (ISO 8601)
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Re: Highest Possible Crit Range?
This is the crux of the issue, because no, you can't "really" get infinite attacks. Mathematically you have an infinite series; while for all intents and purposes you can treat this as being an infinite number, it does not actually exist as infinity.
For the numbers term1nally s1ck mentioned earlier, the odds of getting X number of attacks are going to be approximately 5/9. This is not literally the same thing as an infinite number of attacks, but it does mean that any time you declare you're going to be rolling an attack, you have a 5/9 chance of having however many attacks as you want.
The odds that you will have a thousand attacks to work with? 5/9.
The odds that you will have a million attacks to work with? 5/9.
The odds that you will have a hundred billion attacks to work with? 5/9.
The odds that you will have more attacks than the number of atoms that exist in the universe? 5/9.
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2009-11-29, 04:11 AM (ISO 8601)
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Re: Highest Possible Crit Range?
Well, something *can* be possibly infinity. Just for example, say you got some slew of feats that made it so, with each attack, if you rolled a 10 or more, you get to attack again, with a cumulative +1 bonus. And let's say the designers were silly, and this +1 bonus counted for getting above 10, so if you were on your fifth roll, and rolled a 7, you'd get 7+5=12, giving you another roll and another +1 bonus. There's a certain % chance that you will do this 10 times, giving you a +10 bonus, at which point you will make an infinite amount of attacks. Thus, you can say that each attack has an X% chance of going infinite.
However, that's not what's happening here. For the above to happen, the chance of each attack terminating has to become 0% at some point, which it does not. Instead, this would generate an arbitrarily higher number, so high that it may as well be infinite (maybe this is the aleph-0 you're talking about? might be nice to know what that even is) but it is NOT infinite. It is in fact exactly as far away from infinity as the number 0, or a gaboojillion*, or even negative gaboojillion*.
*A Gaboojillion being X^Xth power X number of times, with X being the number of atoms in the universe5e Homebrew: Death Knight (Class), Kensai (Monk Subclass)Excellent avatar by Elder Tsofu.
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2009-11-29, 04:17 AM (ISO 8601)
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Re: Highest Possible Crit Range?
Aleph-0 is, as far as I am aware, the lowest transfinite number, being defined as the cardinality of the set of natural numbers. It is the largest number which you can name, including by means of this process, with an arbitrary increase in size. It is about as close to infinity as you get while still being vaguely able to work with it sanely.
If I creep into your house in the dead of night and strangle you while you sleep, you probably messed up your grammar.
I'm always extremely careful to hedge myself against absolute statements.
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2009-11-29, 04:22 AM (ISO 8601)
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Re: Highest Possible Crit Range?
That was sexy. I like being shown holes in my argument. *cookie for you* Very interesting, though I think you step-outside the boundaries of mathematics when you start houseruling lol. So the only way you have the ability of possibly achieving the state of infinity is if you can somehow completely eliminate all chance of failure. I dig that.
And Random, I was providing proof that infinity is not a number, but a state. Numbers can function with percentages, states cannot. That's all I was saying with that example. Like I said, your math rocks. I was in no way criticizing or even attempting to understand the sexiness that was those 'math equation bombs'. I was trying to show that infinity was being used wrong, something which didn't really require math to show. That's all.
Oh, and I quoted the differences between Aleph-0 and infinity a few posts back if you are curious.Last edited by Gan The Grey; 2009-11-29 at 04:26 AM.
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2009-11-29, 04:24 AM (ISO 8601)
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Re: Highest Possible Crit Range?
Good. I am glad that we have reached an understanding. You be happy with your proof that it is not quite infinity, and I'll be happy with our proof that it is not quite finite either.
If I creep into your house in the dead of night and strangle you while you sleep, you probably messed up your grammar.
I'm always extremely careful to hedge myself against absolute statements.
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2009-11-29, 04:24 AM (ISO 8601)
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Re: Highest Possible Crit Range?
There will always be a chance for each attack to terminate, but the chance for all attacks to terminate becomes infinitely close to 0 as the number of attacks increases. While you never get a literal infinity of attacks, the number of attacks generated in a successful chain is mathematically equal to infinity.
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2009-11-29, 04:27 AM (ISO 8601)
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Re: Highest Possible Crit Range?
5e Homebrew: Death Knight (Class), Kensai (Monk Subclass)Excellent avatar by Elder Tsofu.
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2009-11-29, 04:29 AM (ISO 8601)
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2009-11-29, 04:42 AM (ISO 8601)
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Re: Highest Possible Crit Range?
Last edited by Wings of Peace; 2009-11-29 at 04:44 AM.
Doc Roc: We're going to eat ourselves.
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2009-11-29, 05:10 AM (ISO 8601)
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2009-11-29, 05:13 AM (ISO 8601)
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Re: Highest Possible Crit Range?
Not quite, when you're dealing with a proof like this...you need a bound, not an approximation, because we don't know how good the approximation is without the bound. In general, if you have an approximation, you can always have it work, because for small/large/specific enough values of your numbers in the equations, you have a bound which satisfies the conditions, but you have to specify what bound you use, because there are a few times when you're dealing with something REALLY close to your approximation that it won't quite work.
By the time it is remotely likely for me to roll that many 1s, I have more attacks than that.
This is here because of the distinction made between deterministically infinite, and non-deterministically infinite.
The posts past here start arguing about aleph-0 vs infinite.
The distinction is that a series (say, the number of attacks) tends towards infinity if and only if it is provable that it eventually exceeds any known natural number. I can't prove that the number of attacks is equal to infinity, because infinity itself isn't really a number. I can prove that the number of attacks tends towards infinity, which is what is known as a 'divergent function' where the limit of f(x) is infinity as x tends to infinity.
Unfortunately, X in this case is the number of attacks made. I have already shown that there is an excellent chance of X finishing in any finite number, so to stretch the whole number line across the 6 seconds we have in the turn, I have to use what is known as the projective real line. In this case, we basically take a line representing 6 seconds, 'twist' them into a quarter-circle, put a point at the centre of the circle it is part of, and draw a straight line down to twice the radius of the quarter-circle directly below the leftmost end of the quarter-circle. This means that as we go through the line, we get higher on the numberline MUCH faster, and we can re-write X as a function of t, and hence f(x) as f(t). But for t = 6, X(6) is defined as infinity, and f(infinity) as the limit of the function, namely infinity. This is a direct result of trying to fit an infinite sequence into a finite time, you HAVE to reach infinity, or you don't have an end-point. The number of attacks made in 6 seconds has a 5/9 chance (or 60/133 for the build I posted) of being infinite.)
This is the non-deterministic infinity.
Aleph-0 is the deterministic infinity, as it is well-defined (the cardinality of the natural numbers), and is NOT the number of attacks I can make, as we're looking at a limit to a sequence, and I'm not sure if I can find a projection from N to the attacks.
(Picky point....aleph-0 is NOT a natural number.)
EDIT: Zeful would be very right...if Infinities weren't a natural result of certain calculations that exclusively involve numbers. How do you think they started to use them? it's not exactly a normal concept....Last edited by term1nally s1ck; 2009-11-29 at 05:27 AM.
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2009-11-29, 05:21 AM (ISO 8601)
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Re: Highest Possible Crit Range?
Ok, now you're just writing down a bunch of nonsense that looks like complicated math but in the end is just a bunch of words. For one, when does the amount of time in a round even come into the equation? It has no relevance to the number of attacks you make at all.
5e Homebrew: Death Knight (Class), Kensai (Monk Subclass)Excellent avatar by Elder Tsofu.
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2009-11-29, 05:30 AM (ISO 8601)
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Re: Highest Possible Crit Range?
By the time it is remotely likely for me to roll that many 1s, I have more attacks than that.Last edited by Gan The Grey; 2009-11-29 at 05:31 AM.
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2009-11-29, 05:37 AM (ISO 8601)
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Re: Highest Possible Crit Range?
Forgive me, I was under the impression that we were trying to consider the amount of attacks made in a single round. We can't count across the attacks themselves, because there is a non-0 chance of them never ending. There's only the two ways to sum across a range with no limit, one is the sum to infinity (which people don't like, because it has the word infinity in it...despite that you sum it as you would sum any finite sum (unless you actually do the finite sum 1-by-1....), or by the projective number line. For the projective number line (google it if you don't want to take my word, it's pretty easy to understand), you can use any variable you want for the semi-circle , provided there's a way to spread the original variable out across the one used. Time was just an obvious one to me, because it's a bounded variable such that all the attacks happen within the bounds. I've now used both, both give an answer of non-finite and infinite, respectively....the two are identical.
if you want me to formalise my above post, tell me...but the arguments are such that I'll have to use the logic symbols, and I still just call on the endpoint being equivalent to infinity, because that's the definition of the projective representation.
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2009-11-29, 07:57 AM (ISO 8601)
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Re: Highest Possible Crit Range?
None of which explains how you get over the fact that, even if you're rolling an arbitrarily high number of attacks, you can't then roll an arbitrarily large number of 1s in a row. Since this will happen literally an infinite number of times before you reach infinity, it has a 100% chance of occurring before then. How small the chance is is completely irrelevant. If it's there, it will happen.
5e Homebrew: Death Knight (Class), Kensai (Monk Subclass)Excellent avatar by Elder Tsofu.
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2009-11-29, 08:15 AM (ISO 8601)
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Re: Highest Possible Crit Range?
Because 'Arbitrarily high' means any finite number you care to name is eventually rolled. Yes, I eventually roll 1 quadrillion nat 1s. By that time, I have more than 1 quadrillion attacks, almost guaranteed. (expected number of attack rolls required to roll 1 quadrillion (10^15) nat 1s = 2^1000000000000000, expected number of attacks at that point = 1.1(or whatever your build happnes to generate per attack)*(10^15), which is MORE than 10^15.
Just because you can roll an infinite number of natural ones doesn't mean I don't have more attacks...after all, there are an infinite number of odd numbers, and yet if I remove them all from the natural numbers, not only do I have some left, but I have infinitely many left.Last edited by term1nally s1ck; 2009-11-29 at 08:20 AM.
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2009-11-29, 08:39 AM (ISO 8601)
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Re: Highest Possible Crit Range?
Yes, it's possible that you won't get enough 1s for the chain to end for an absurdly long time, but you're severely underestimating infinity. It does not matter how small the chance is that the chain will end, if it's more than zero, it WILL END. PERIOD. Infinity doesn't care if it keeps becoming less and less likely that the chain will end, all it cares about is whether that value is zero, or not zero. Since it's no zero, it happens.
5e Homebrew: Death Knight (Class), Kensai (Monk Subclass)Excellent avatar by Elder Tsofu.
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2009-11-29, 12:58 PM (ISO 8601)
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Re: Highest Possible Crit Range?
I feel that we have the same problem as before. We have a non-rigorous conceptual agument being used to try to shoot down a rigorous mathematical proof. I suggest that you either attack that proof mathematically, or admit defeat.
If I creep into your house in the dead of night and strangle you while you sleep, you probably messed up your grammar.
I'm always extremely careful to hedge myself against absolute statements.
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2009-11-29, 03:18 PM (ISO 8601)
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Re: Highest Possible Crit Range?
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.