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Everything posted by uncool

  1. 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.
  2. 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.
  3. 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
  4. 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.
  5. 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.
  6. 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.
  7. 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.
  8. 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.
  9. 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.
  10. 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.
  11. 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.
  12. I expect that you should be able to look something up when explicitly told its name. https://en.wikipedia.org/wiki/Law_of_large_numbers "In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed."
  13. As I said earlier: If you don't mean the law of large numbers, then I still can't discern a precise statement you think is false.
  14. Can you make a precise statement that you think is false?
  15. And why, according to you, was there a drop in the intensity of indoctrination of children?
  16. And Ghideon is telling you that you are asking for a proof of something false. As you flip an increasing number of times, the absolute difference between the number of heads and tails will tend to grow, but the ratio of heads will tend to 1/2.
  17. ...that's pretty much what Ghideon said.
  18. Do you understand what "absolute difference" means?
  19. That's not an answer to the very specific statement Ghideon made.
  20. He is correct; he is talking about the absolute difference between the number of heads and tails. Only the ratio will approach 1/2.
  21. It sounds like you're asking about the law of large numbers, a theorem known for binary variables since 1713. This is too vague to even mean something. Proof is for specific statements. What is the specific statement that you are asking for proof of? This isn't true, actually. They would find that the ratio of heads would approach 1/2, but that the number of excess heads would diverge from 0.
  22. What "problem"? Probability theory and statistics have been a branch of mathematics since the 1600s, if you start counting with Pascal and Fermat. I have no idea what you are trying to ask here.
  23. ...am I the only person to notice which subforum this is in?
  24. There is no sense in repeating the bare assertion, no. If you want to convince us that your assertion is correct, you can write the proof to your assertion - without handwaving. If you want to show that it might be correct, you could demonstrate that wtf's order isn't a counterexample without repeating the assertion as if it were proven fact. But simply repeating it is a waste of your time and ours, yes.
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