uncool

It doesn't directly, because the law of large numbers isn't about events occurring in a row. If you mean does it consider the possibility (by including it in probability calculations), then yes, it does - the possibility is included in the probability space.

From Wikipedia: " A special form of the LLN (for a binary random variable) was first proved by Jacob Bernoulli.[2] It took him over 20 years to develop a sufficiently rigorous mathematical proof which was published in his Ars Conjectandi (The Art of Conjecturing) in 1713. He named this his "Golden Theorem" but it became generally known as "Bernoulli's Theorem". " The actual proof (using modern terminology and notation) of the weak law of large numbers is also in the article: https://en.wikipedia.org/wiki/Law_of_large_numbers#Proof_of_the_weak_law

P(H or T) means the probability of getting a result that is either heads or tails. In other words, if the flip is heads, or if it's tails, it's accepted. Which means any result (other than "landing on the side") is accepted, so P(H or T) = 1. What I'm asking you for is basic probability that has been known for literal centuries. It's a generalization of the following question: What is the probability that after 4 flips of a fair coin, the number of heads is between 2 and 4, inclusive? (The answer will be 11/16) To be blunt: you are attempting to make pronouncements when you don't know the basics of this field of mathematics. You don't know what you are talking about, and are striking out randomly, resulting in exactly what I said before: quasi-randomly putting formulae together without understanding what they are for Not knowing probability theory isn't a problem. Everyone was there at some point. The problem is that you don't know probability theory, but you are confidently making assertions about it anyway. Probability theory is a very well-studied area of mathematics, and has been a part of it for centuries. The law of large numbers is a basic theorem in probability, and has been mathematically proven, again, for centuries.

You are making my point. Why are you looking at the probability of getting a heads or tails (which will be 1), when what I asked you for was "the probability that after 2000 flips of a fair coin, the number of heads will be between 990 and 1010"?

That's still not an answer, and worse, it is notationally nonsense. What do you mean by P(H or T)? As I said before:

Yet again, that not only doesn't contradict what I said, it means the precise same thing: "And the theorem says that the total probability of landing within that range limits to 1." "The proportion of heads after n flips" is not "the total probability of landing within that range". And? I still don't see an answer for the probability that after 2000 flips of a fair coin, the number of heads will be between 990 and 1010.

Once again: what does "it" refer to here? Hahaha whatever you say ...this is nowhere near the formula I asked for. To demonstrate: please show me how you would use this formula to calculate the probability that after 2000 flips of a fair coin, the number of heads will be between 990 and 1010.

Yes, and that doesn't contradict the statement I made: "And the theorem says that the total probability of landing within that range limits to 1." (In fact, it says exactly the same thing) No, and that's not even close to what I said. Caught up in what? The fact that the theorem deals with a range, and not only "perfect" outcomes? Because you keep on talking only about the "perfect" outcomes. You did not (or at least, did not do so correctly). If you had, you would have gotten the formula I stated. Here's an exercise for you: write a formula for the probability that after 2000 flips of a fair coin, the number of heads will be between 990 and 1010.

"No, the law of large numbers approaches the expected value which is 1/2" makes no sense. That isn't a definition, it's a garbled version of the law of large numbers itself. And it doesn't address my point: as the number of flips increases, the range of "acceptable" outcomes also increases. Which means you don't only calculate the probability of the "perfect" outcome (i.e. the closest number of flips to 1/2), but of an entire range of numbers of heads.

What does "it" refer to here? "the total probability of landing within that range"?

But you would also accept 101 heads and 100 tails, right? And if you flipped 2001 times, you would probably accept anywhere between 991 and 1010 heads as evidence of a fair coin. And if you flipped 20001 times, you would probably accept anywhere between 9901 and 10100 heads. And so on. So the question isn't about the probability of a specific number of heads, but a range that grows with the number of flips. And the theorem says that the total probability of landing within that range limits to 1.

That (or at least, some reasonable version of that: take the sum of the products of the probabilities and the values) is the definition of the expected value of a random variable.

The law of large numbers has been proven mathematically. I could provide a basic proof, if you want. Please explain what you mean.

It has been proven. The law of large numbers is a theorem. I have no idea what you mean by this. "I don't see how" doesn't mean it can't be done.

This is not precise. I am asking you to be precise. The integral of what function will be equal to what sum? I think you are likely misremembering how Riemann sums are used. That is the method to get the expected value, yes. Yes, which doesn't say anything about, for example, what happens when you roll 4 times. First, I did define a and b (in an earlier post): "For any a < 1/2 < b". Pick any a and b satisfying those inequalities, and the limit I wrote will be correct. Second, I guarantee that it is not, in part because the summation is over values of i, not over values of n, in part because the "division by n" is about the bounds (the sum could equally be written as "sum_{a < i/n < b}"), not the value, and in part because there is no integration happening.

Please explain precisely how you are using the fundamental theorem of calculus here. The law of large numbers is about a limit, so no, what happens for small numbers does not affect it. The formula you have written is not what I gave you. I told you the exact formula: lim_{n -> infinity} (sum_{an < i < bn} (nCi)/2^n) = 1

Which "law of calculus"? Please explain specifically. The weak law is still about large numbers. That still doesn't explain the formula you are using.

That is a coincidence. If you had 6 rolls, there are 64 sequences, not 216. The above provides no justification for integrating anything, and reads like what I said above: "quasi-randomly putting formulae together without understanding what they are for, rather than figuring out precisely what you want, then determining the formula for that." I have no idea what you are saying here; the law of large numbers says nothing about what happens for a specific number of flips - it's about large numbers. Again: why integrate? What is this formula supposed to calculate? And the 1/n restricts "allowed outcomes"; it doesn't multiply any probabilities.

It should be nCr/2^n, not nCr/n^r, if by "desirable outcome" you mean "exactly r heads". Integral with respect to what, and why is the integral the appropriate choice here - especially when everything is discrete, not continuous? Err, 1/2! = 1/2, not 1. The mathematical statement of the theorem as applied to this case would be: lim_{n -> infinity} (sum_{an < i < bn} (nCi)/2^n) = 1 Summation, not integration, over possible numbers of heads, not over n.

1) The limit you have set up doesn't make sense, because you quantify n twice. 2) It looks to me like you are quasi-randomly putting formulae together without understanding what they are for, rather than figuring out precisely what you want, then determining the formula for that. What, exactly, is that formula supposed to calculate? 3) Yet again, "getting the same number of heads and tails" is not the right way to look at it. 4) Is your post supposed to be a response to mine? It doesn't refer to anything I actually said... The precise statement of the theorem for this case is: For any a < 1/2 < b, for any epsilon > 0, there exists some N such that for any n > N, the number of sequences of length n with number of heads between a*n and b*n is greater than (1 - epsilon)*2^n.

By allowing any (reasonable) choice of what "close to" means, using quantifiers. The theorem says that for any reasonable choice of "close to" (read: for any open interval containing 1/2), the proportion of outcomes covered by that "close to" will get arbitrarily close to 1 (as the number of flips increases). In short, by using nearly the same setup as standard limits.

You are missing my point. Again: it isn't only about "outcomes with the expected value". It's about outcomes close to the expected value. For a quick (and imprecise) example: Consider the sequence H, H, T, H, T, H, T, ... At no point is the number of heads equal to the number of tails. But the average number of heads is still 1/2.

Ironically, I am severely skeptical of this idea.

Because for any range around the expected value of 1/2, say, 0.498 to 0.502, the number of outcomes (i.e. number of "allowed heads") within that range increases as N gets larger. In other words, the sequences that "help" the theorem (speaking very, very non-precisely) do not have to be only those that get exactly the same number of heads and tails.

I get that you're trying to talk about something close to the law of large numbers (which is why i referred to it), but still don't understand what the "problem" is supposed to be, no.