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uncool

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Everything posted by uncool

  1. It is not contradictory, because the range is growing. For 200 flips, the range is between 99 and 101 heads - 3 outcomes. For 2000 flips, it's between 990 and 1010 - 21 outcomes. For 20000 flips, it's between 9900 and 10100 flips - 201 outcomes. For 200 flips, the probability of exactly 100 heads is about 5.6%. For 2000 flips, the probability of exactly 1000 heads is about 1.8%. For 20000 flips, the probability of exactly 10000 heads is about 0.56%. The probability of getting exactly the "perfect" number of heads is decreasing, but the probability of hitting a range including 1/2 is increasing.
  2. I've demonstrated an example of the law of large numbers (ish; technically, all I've done is point out some increasing values, but careful investigation will continue to show that the point I make with those values is correct), in direct contradiction to one of your statements. It warrants you retracting that statement, as a start.
  3. And as defined by the law of large numbers, it grows with n. Which is what I showed. It sounds like you disagree with the axiom that the probability of a union of disjoint events is the sum of the probability of each of those events. Actually, it does change that. If you flip a coin 200 times, the probability of (number of heads/number of flips) being between 0.495 and 0.505 is about 17%. If you flip a coin 2000 times, the probability of (number of heads/number of flips) being between 0.495 and 0.505 is about 36%. If you flip a coin 20000 times, the probability of (number of heads/number of flips) being between 0.495 and 0.505 is about 84%. Because the law of large numbers is about that range around "perfect".
  4. I am demonstrating that the weak law of large numbers talks about an increasing range of "accepted" outcomes as n grows larger. Which is what you asked me about.
  5. I'm not. I'm applying probability theory. It's not. Don't. It means what it says, and implies exactly what I said it did.
  6. "But instead of taking the sum of a discrete number of things, you're taking the sum of an infinite number". We're looking at discrete numbers of heads. That comes into play because of the expression. |bar(X)_n - \mu| > \epsilon is equivalent to |(X_1 + X_2 + ... + X_n)/n - \mu| > \epsilon is equivalent to |X_1 + X_2 + ... + X_n - n \mu| > n * \epsilon The right-hand side grows with n. It says that the sum can be n*epsilon away from n* mean.
  7. In which case you use a sum, because the outcomes are discrete. As I have repeatedly shown. In the expression "converge in probability" (for the weak law, which is all that is necessary here). The "Pr" in "lim_{n -> infinity} Pr(...) = 0" denotes probability.
  8. This is what I mean by "quasi-randomly putting formulae together". There is no reason to be integrating in the first place. You didn't read my answer closely. The law of large numbers is entirely about probabilities. It's a theorem in probability theory. It talks about convergence in probability.
  9. This is the number of sequences of n flips such that r of them are heads. The "possible outcomes" are sequences of n flips; the number is 2^n. The number of outcomes does not depend on r. So you take a sum. Not an integral, a sum. Further, the sum is over r, not over n. sum_{range of r} nCr/2^n Further, there is no reason to be taking a limit at this point. No, you already have a probability. The division by n was to get an average value, not an average probability.
  10. @studiotIf anything, I see it the other way around. The law of large numbers is what Conjurer has referred to over and over (including in the OP, with "the outcome of flipping more and more coins in a row approaches closer and closer to half of the coins being heads or tails."); if anything, I see "n becoming transfinite" as a diversion. What do you mean by "the way we calculate probabilities"? Do you mean, for example, the axiom that the probabilities of disjoint events should add?
  11. And your concern is misplaced, as shown by the law of large numbers itself (which is, I repeat, a proven theorem).
  12. If what you mean is that there's always a chance of getting it wrong, of a fair coin coming up heads, heads, heads, heads, etc., then yes, that's how probabilities work. There's a chance. But it's really unlikely, and as the number of flips gets higher, the outcome becomes more and more unlikely. And that has very little to do with some of the things you were saying earlier. Otherwise, I still have no idea what you are trying to say.
  13. 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.
  14. 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
  15. 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.
  16. 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"?
  17. 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:
  18. 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.
  19. 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.
  20. 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.
  21. "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.
  22. What does "it" refer to here? "the total probability of landing within that range"?
  23. 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.
  24. 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.
  25. The law of large numbers has been proven mathematically. I could provide a basic proof, if you want. Please explain what you mean.
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