Hi,

I think i agree except do we really need to state 2*inf because that's just infinity right?

Possibly. I specified it in that form so no one would say “What about an odd size. And it didn’t matter because as you say 2*∞ is the same as ∞ for the discussion. (of course, in general it depends on what you mean by “just infinity”. Infinities are not necessarily the same)

 

Hi,

Apparently there are a number of misunderstandings going on in this discussion.
For one, you dont seem to get that infinity is NOT a large number. The only time a large number can represent infinity is when the result of some calculation drops off fast not when it drops off gradually. I think this current test shows a gradual drop off not an exopnential drop off so the only thing that probably works is infinity and nothing short of that. Not 10^10, not 10^100, not 10^1000, not a google of zeros after a 1.
So there is no finite number you can choose that will work unless we can show that the error drops off exponentially because if it does not then there will always be an error until the theoretical infinity is reached, and that of course is not possible except in theory.

So if your test is based on something that can only be true for exactly infinity and nothing less, then it's not a very practical test, is it?

You are basing your test on the expected results approaching what you would expect for the case of infinite trials but now you are claiming that the behavior of the system as the number of trials grows without bound is completely and radically different than it would be for the case of infinite trials. You can't have it both ways.

Let's see if I can rephrase this to get at the heart of the matter (or one of them) more explicitly.

I flip a coin (an ideal fair coin) 2N times and I get H heads and T tails. I then calculate X = H - T and I consider this trial a success if the magnitude of X -- call this the spread -- is no more than 100. In other words, it's a successful trial if the spread between the number of heads and the number of tails is no more than 100. Does the probability of success increase of decrease as N gets larger and larger?

Put simply, does the expected spread increase or decrease as the number of flips increases?

 

Possibly. I specified it in that form so no one would say “What about an odd size. And it didn’t matter because as you say 2*∞ is the same as ∞ for the discussion. (of course, in general it depends on what you mean by “just infinity”. Infinities are not necessarily the same)

Hi,

Yeah maybe we could look into this. I like your rationale for 2x inf.
Mine is that if we had such a large large (inf) number of samples then 1 more can only change the sum by 1/inf which is really zero.
For example, with 200 samples one more 'heads' would mean 101/100 or 100/101, and with 2000 samples one ore one mean 2001/2000 or 2000/2001, and as we increase N this number gets closer and closer to 1 even though we now have an odd number of samples.

 

So if your test is based on something that can only be true for exactly infinity and nothing less, then it's not a very practical test, is it?

Well not really, the test you seem to be doing requires N going to infinity, but the test i am doing only has to tend to a large number. What we are doing is looking for a trend and that comes after several increasingly large N experiments. We see the error get lower in general although we never can see it actually go to zero except in theory.

You are basing your test on the expected results approaching what you would expect for the case of infinite trials but now you are claiming that the behavior of the system as the number of trials grows without bound is completely and radically different than it would be for the case of infinite trials. You can't have it both ways.

I dont understand why you are saying i cant have it both ways. If there are two behaviors, there are two behaviors, and one follows logically from the other from what i can see. These infinity problems often work like this but i have seen two cases, 1 where the error goes really fast toward zero and 2 where the error goes slowly to zero but nonetheless does get smaller and smaller. We seem to see the latter here but at infinity i believe it should be zero and this might be easy to prove for the coin test.

Let's see if I can rephrase this to get at the heart of the matter (or one of them) more explicitly.

I flip a coin (an ideal fair coin) 2N times and I get H heads and T tails. I then calculate X = H - T and I consider this trial a success if the magnitude of X -- call this the spread -- is no more than 100. In other words, it's a successful trial if the spread between the number of heads and the number of tails is no more than 100. Does the probability of success increase of decrease as N gets larger and larger?

Put simply, does the expected spread increase or decrease as the number of flips increases?

I dont think you can look at it that way because even if the 'spread' increases the average might still get closer and closer to the expected mean of 1 and 0 which is 1/2. That's because the sum gets larger and if it gets larger faster than the spread then the average is tending toward 1/2. I think that's right but i could test it i guess.

But at this point i have to ask you what did the tests i already did say to you? Especially the 3rd one which i had shown first in the former post?
What we see goes somewhat like this:
we might see (H/T) 10/20 which means the mean is 1/3 which is not one half, but as N gets larger we start to see something like 150/200 which is closer to 1/2, then as N gets larger we see something like 1800/2000, then maybe 19000/20000, then maybe 199000/200000 and we might see then 1999999/2000000 and so we see the mean getting closer and closer to 1/2.
What does that tell you?
We can also see it bounce: first 10/20 then 100/90 but that's still closer to a mean of 1/2.
We can also see it go higher for one run then back down again on the following run but overall it gets closer to 1/2.

So what does that numerical test say to you?