Highest Possible Crit Range?

[randomaeiou]

ummm... yes, i understand what you guys are saying, and accept that argument.

playing devil's advocate:

in this particular case, you accumulate attacks at a FAR faster rate than runs of 1's are likely to occur, so the expectations are realistic.

now, applying your argument to a slightly different situation: imagine you accumulate attacks at a rate just above (approaching) one over the rate of runs of 1's. to infinity, that is essentially irrelevant, since you ARE accumulating attacks and will therefore have a buffer, hence EXPECT to go infinity. but thinking about it, if you had to roll a d100 in the same situation and only got a single extra attack on 50 and above (i.e. 51 vs 49), we would all agree infinity is really rather unlikely.

Based on signmaker's argument about "vanishingly small", there seems to be a point at which we accept the argument and says it goes to infinity despite the chance of it not doing so. I understand 51vs49 being hardly "vanishingly small", but at what point does it do so? At what point can we call such a series infinite? (This is a theoretical exercise, but has applications in any system where positive feedback is expected, such as nuclear reactions...)

Based on Kalirren's last paragraph - intuitively, there is a certain amount of attacks which you need to have stored up before we call a run of 1's terminating the sequence "really unlucky to the point of improbability within an infinite series". What would be this point?

One might argue the relevance/importance of this point, but it does have implications - just how little fissile material do you need to build a runaway nuclear reaction, for example, or conversely, how much fuel can you safely put in a reactor so that it will naturally terminate if unchecked, rather than go Chernobyl on everyone...

[term1nally s1ck]

The problem you have, is when we say we're expected to reach infinity, we don't mean it's guaranteed.

In fact, in the case of 51% vs 49%, the chances of the sequence never terminating are very very small...2/51 in the limit, to be precise.

However, your expected number of attacks is then at least 2/51 * infinity, which is still an infinite number.

The point where you give up and just assume it is infinite is when you have enough attacks that the probability of the sequence ending is small enough that you don't care about it anymore. (1 in a billion or so, for example.) To do that for the original build, which has an 8/11 chance of reaching infinity from a single attack, you'd need to garner 14 attacks. I'm not going to even bother to calculate how many you'd need to get to that for the 49-51 example...it's a hell of a lot.

The only time you no longer have a chance of the sequence not terminating is when you reach a 50-50 chance.

[randomaeiou]

In fact, in the case of 51% vs 49%, the chances of the sequence never terminating are very very small...2/51 in the limit, to be precise.

However, your expected number of attacks is then at least 2/51 * infinity, which is still an infinite number.

Taking those two statements together, which is it? So if I have crit range 10-20, (assuming to hit is below that, and confirming crit as auto to simplify things), does that mean I do infinite damage, since my expected number of attacks will be 2/51*infinity?

Clearly that doesn't make intuitive sense, but based on your second statement...

Also, there is one concept bugging me about all this. Within infinity, do all possibilities occur? If it does, then basically, if something CAN occur, it WILL (probability thereof regardless). I understand the "vanishingly small" argument about not expecting the sequence to terminate, although it WILL within infinity (ignoring the minute but nonzero possibility does not make it go away!) - but still, if it DOES occur within infinity, as it has to, since we are talking about ALL possibilities, does that make it NOT infinite then?

Basically, ignoring minute/irrelevant possibilities is typically an engineer's way of thinking, whereas the absolutes tends to be how mathmos+scientists behave, from personal experience. I know we are arguing about a possibility so infinitesmal as to be totally irrelevant, and any self-respecting engineer would have rounded it off and dismissed it about 10 pages back. But this is a maths question, not one of "what works", but "what is". Heck, that was how Einstein triumphed over Newton (with Eddington's starlight photos - how relevant is that?!)

Basically, in my head, infinity makes probability irrelevant - only absolutes affect an outcome in an infinite situation. If it can happen, it will - so if you cannot guarantee that our series will continue indefinitely, then it wouldn't, in infinity. Only if it is *impossible*, then does it not have an effect on an infinite series, n'est-ce pas?

[term1nally s1ck]

It's Both.

The Expected number of attacks is the average number of attacks you'd make if you ran through various series of attacks repeatedly.

As an example, the Expected number of heads you'd get if you flip a coin 10 times is 5. That doesn't mean that you always get 5, or that it's even likely that you get 5 (the chance of getting 5 exactly is actually quite small, under 1/4)

So yes, on average, you do infinite damage. You don't do infinite damage every time you start attacking, or even regularly, but because you pull it off sometimes, and because of the way averages work, that makes your average damage infinite.

You argument over things being guaranteed within infinity is limited because of one thing. Only things with positive probability are guaranteed to occur. An infinite string of misses has probability 0, so is impossible, and doesn't happen except possibly at infinity. Any finite string of misses (any number you can write down, or imagine) IS guaranteed, but by the time they're going to occur, you have far more attacks than that, so you carry on attacking anyway.

A Mathmo still deals with infinitely small numbers, and still dismisses them (if you look at Newton's calculus, the basis of almost all mathematics we use nowadays, the major step is looking at what happens as a very very small value tends to 0, and dismissing anything that tends to 0 with it as infinitesmal.), but only if he can show without doubt that those values have no effect in the long term. I've shown a couple of times that in the long term, it's provable that the total probability of failure in finite time is not a guarantee, meaning that there is at least a positive probability of reaching infinity.

Of course, you can use your argument that everything occurs at infinity, to decide that yes, the string of attacks does end at infinity. But then it has to get there first.

[Heliomance]

Infinity breaks intuition. Also, the expected value of a system is badly named. The expected number of attacks is not actually the number of attacks you expect to make. What it actually represents is, if you were running the experiment an infinite number of times, what's the average number of attacks you'd make in each experiment. So just because the expected value is infinity, it doesn't mean that in a given experiment you expect to make infinite attacks. It means that on average, given infinite experiments, you will have infinite attacks.

[randomaeiou]

So yes, on average, you do infinite damage. You don't do infinite damage every time you start attacking, or even regularly, but because you pull it off sometimes, and because of the way averages work, that makes your average damage infinite.

But if even a single rather improbable sequence can lead to infinite damage, does that not make the average damage of that particular series infinite? (e.g. coin flipping and getting infinite heads - rather improbable, but since it only takes a single instance of an infinite number of heads to average out everything to infinite heads... and extremely counterintuitive!) (BTW, does infinity divided by infinity = 1, or does the world ex(im?)plode at that equation?)

You argument over things being guaranteed within infinity is limited because of one thing. Only things with positive probability are guaranteed to occur. An infinite string of misses has probability 0, so is impossible, and doesn't happen except possibly at infinity. Any finite string of misses (any number you can write down, or imagine) IS guaranteed, but by the time they're going to occur, you have far more attacks than that, so you carry on attacking anyway.

Of course, you can use your argument that everything occurs at infinity, to decide that yes, the string of attacks does end at infinity. But then it has to get there first.

How about I imagine a finite number of misses which is exactly equal to the finite number of attacks that are left - it's certainly positive probability of occurance... (That is the crux of my argument - that the number of attacks is some finite number until it truly reaches infinity, and along the way whilst it is still finite, one can imagine a finite string of misses which will cause the series to whiff. As long as this is possible, it will cause infinity to not happen, and therefore this series is not infinite, just near it? remember - probability in the case of infinity is irrelevant, just possibility. I accept that it is a vanishingly small probability, but infinity will be broken just by one instance of it - which you cannot guarantee will not happen. In fact, you can guarantee WILL happen since to get to infinity you have to have gone through ALL possibilities...)

Hmm... to rephrase our problem as succintly as possible:

In a biased random walk with a positive but <100% probability of going infinite, is the result finite or infinite?

[term1nally s1ck]

But if even a single rather improbable sequence can lead to infinite damage, does that not make the average damage of that particular series infinite? (e.g. coin flipping and getting infinite heads - rather improbable, but since it only takes a single instance of an infinite number of heads to average out everything to infinite heads... and extremely counterintuitive!) (BTW, does infinity divided by infinity = 1, or does the world ex(im?)plode at that equation?)

Since the probability of an infinite string of heads is 0, you're looking for something which doesn't happen. The probability of it happening in this case is not 0, so we get an expected value of 0.

How about I imagine a finite number of misses which is exactly equal to the finite number of attacks that are left - it's certainly positive probability of occurance... (That is the crux of my argument - that the number of attacks is some finite number until it truly reaches infinity, and along the way whilst it is still finite, one can imagine a finite string of misses which will cause the series to whiff. As long as this is possible, it will cause infinity to not happen, and therefore this series is not infinite, just near it? remember - probability in the case of infinity is irrelevant, just possibility. I accept that it is a vanishingly small probability, but infinity will be broken just by one instance of it - which you cannot guarantee will not happen. In fact, you can guarantee WILL happen since to get to infinity you have to have gone through ALL possibilities...)

Yeah, it's a positive probability, but it takes more and more misses to stop the sequence as time goes by, so the probability gets even smaller the longer it lasts. So yes, on the way to infinity, we'll go through every instance of a string of misses of every possible length. However, because of how quickly we gain extra attacks, the chances are (and we CAN apply probability to the infinite case, by taking the limit of the finite sum over all possibilities as the upper bound tends to infinity) that we will have more attacks than that particular string of misses every time we hit one.

No, I can't guarantee that the sequence will not terminate. What I can guarantee is that if you run the series of attacks a large number of times, roughly 8/11 of the sequences of attacks will not terminate in a finite length.

If we reach infinity, your argument applies perfectly, and we can assume that a sequence of misses long enough to end the attacks will happen. However, at that point, we've already reached infinity, and this sequence is said to 'terminate at infinity'.

You're unfortunately trying to apply some of the intuitive proerties of infinity to the sequence before it reaches infinity, which cannot work because you aren't there yet.


EDIT: Your last question there reveals the problem. In any given random walk, the result is what it is. It could end up at 0, or it could end up increasing without bound, and never getting back to 0 in finite time. I'm not claiming that it always reaches infinity, not even close. I'm claiming that there is a significant chance that it reaches infinity.