Author: fca
Date: 06:17:26 08/17/98
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On August 17, 1998 at 06:09:24, Dan Homan wrote: >On August 14, 1998 at 21:48:40, fca wrote: > >>Just fill in the blanks for me. It gets better as it goes along. All are of >>course "answerable". And all are welcome to reply. Dan, Bruce and Dann >>especially. > >Ok, I'll bite. There is probably a catch here somewhere. :) If not, it would be inhospitable of me. >>(a) In 1048576 trials (each of 20 throws) the probability of one or more "20 in >>a row"s occurring is ________ . > >Assuming that we don't care whether it is 20 in a row of heads or 20 in a row >of tails, I was confident you would not stumble here. Well spotted. However, there is another ambiguity. The trials were not stated to be non-intersecting. In fact, the simplest trial definition would be any 20 consecutive throws. Dann Corbitt spotted this catch. Your answers for (a) and (b) are of course perfectly correct if trials are assumed to be consecutive and non-intersecting, not that you needed me to tell you this. Could you, Dan, and anyone else, therefore please rework (a) and (b) with the following clarifications (both conditions must apply): (1) Interpretation X: A trial is any 20 consecutive throws, and trials are numbered sensibly (so Trial 1 and Trial 3 have 18 throws in common, for example); ***AND*** (2) Interpretation Y: "20 in a row" means 20 heads OR 20 tails in a row. I save you from the bother of working it out _just_ for 20 heads... I am sure (2) may give you a little more fun instead... ;-)) [correct - barring the above clarification - computations snipped] >>(c) In 1048576 such trials, where exactly 4 "20 in a row"s were observed, there >>is a _______ % chance that this result does not contradict the "fair coin" >>hypothesis. > > P(4 "20 in a rows" in N trials) = > (N!/(4!(N-4)!)) x (2 x (1/2)^10)^4 x (1 - 2 x (1/2)^20)^(N-4) > >Again, assuming we don't care whether the "20 in a row" is of tails or heads. > >I will leave the above as an exercise to the reader, if N = 1048576. :) Why? Stirling's formula for big factorials not good enough?! ;-) And, does not your answer give the probability of 4 "any 20's"? Rather than the probability that the result does not contradict the initial "fair coin" hypothesis?? Say the P you computed above = 0.2 ... surely you are not then saying the probability that the result contradicts the fair coin hypothesis is 0.8? Or that the probability that the result does not contradict the fair coin hypothesis is 0.2 (if this statement is different from the last statement)? I urge reconsideration.... you can take Interpretation X or the alternative (Trial 2 begins on throw 21), whichever you consider easier (if different). We'll assume Interpretation Y of course applies. >>(d) "20 in a row" has been just been thrown for the first time in the >>experiment. The best guess of the trial number on which this happened is >>_____________ . >It is, of course, equally likely to happen on any of the trials, Have you taken full account of the fact that this is the "first" occurrence? Bruce certainly did... Is not the modal value the first one, thereby, as per Bruce? > but the >expected number of trials should be Was that the question? Are you sure? ;-) Continuing as if it was the question (d)... > something like.... :-) >N = 1/(2 x (1/2)^20) = 524384 >Again, assuming that we don't care where the "20 in a row" is of tails or >heads. Note that the above expression is slightly incorrect, the denomentator >should also include the chance of not having 20 in a row - but this >is very close to 1. We are dealing with a big number as a final answer to the computation though, and the m < < M of physics may be applied here only with caution... Of course I noted the "something like" :-) > - Dan >P.S. If you want the results for just "20 heads in a row", simply change >the 2 x (1/2)^20 is all of the above expressions to (1/2)^20. Until now, perhaps. But with Interpretation Y as well, do you maintain that is still the case? When trial one ended with ...HT, Trial 2 ain't going to be either 20 heads or 20 tails - definite. ;-) Now you have bitten, Don, please do not abandon. [reeling-in mode engaged, perhaps] :-) Please, no one should misunderstand either the goodwill underpinning _all_ the posts in this thread, or the chess-relevance of (c) and (d) at least. Bruce, Don, Donn.... please help ;-) Kind regards fca "In Physics, Vigour Is A Virtue. In Mathematics, The Converse May Be True, And Instead Rigour Is A Pre-Requisite" As applied to chess of course... "The Universe, Maths And Chess Are Not Just Stranger Than One Might Imagine. They Are All Stranger Than One *COULD* Imagine" Both quotes originals by fca, the latter adapted from a well-known one. PS: I hope I have not spoiled Donn Corbitt's holiday with these infuriating questions. Donn picked up Interpretation X pretty immediately...
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