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If I look at a sequence of 100 coin tosses and I get 90 heads the probability of the coin being fair vs not fair is a lot lower as answered in a lot of questions already. But when I go from looking at 90 heads to the actual permutation of heads and tails, that permutation is equally as likely to be caused by a fair coin as any other permutation. So that permutation being equally as likely as any other how could we say that the coin being fair is less likely?

Am I confusing the direction of the statements? A fair coin is equally likely to cause this permutation, but the permutation is less likely to be caused by a fair coin? This seems weird to say and I have trouble unifying the two views. I'm making some kind of logical error.

mdewey
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Siniyas
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    I *think* what you're asking is that since all combinations of $100$ outcomes are equally likely for a fair coin, why should observing 90 heads be any more evidence against the hypothesis of fairness than observing 50, 51, or even 100 heads? If that's correct, then the answer lies in the theory of hypothesis testing: see http://stats.stackexchange.com/questions/31. – whuber Sep 26 '16 at 14:46
  • I understand, that if you say observing 90 heads is less likely than observing 50 heads, because the set of permutations for 50 heads is by far greater. But if we make an experiment of throwing a coin 1000 times and it comes up heads 900 times we can not say that a fair coin is less likely to cause that permutation than any other. Yet seeing that result we would doubt that the coin is fair. I think that is because while the probability of the fair coin causing the permutation is the same, a coin that had a 0.9 prob of landing on heads has a much higher chance of causing this permutation. – Siniyas Sep 27 '16 at 12:27
  • "we can not say that a fair coin is less likely to cause that permutation than any other". But that's not what you're asking. You're asking if that permutation is as likely to be caused by a different coin. Write out the Bayes rule and reason with it. – Neil G May 23 '17 at 07:33
  • What do you mean by permutations? Do you ask if there is a difference in HHTTHT vs HHHTTT permutations? No, both are equally likely since coin does not care about previous outcomes. – Tim May 23 '17 at 08:24

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