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I have a question I've been wondering about.

Say you have some weighted coin and you have a reasonable belief that it has a 1/10 chance of heads on a flip (although this is not necessarily the true probability).

And after 5000 trials you obtain 455 successes.

Is there any particular method that would be preferred for a) testing against the null hypothesis P=0.1 and b) estimating the true probability of heads on each flip? Would a Bayesian approach be significantly different than just doing a binomial test? Thanks in advance!

user1133454
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  • Testing and estimating the probability are two problems. Which one do you want to do? – Tim Dec 22 '22 at 22:04
  • Estimate the true probability, I suppose. I know how to make the confidence interval, but I’m wondering about what other approaches exist for this problem. – user1133454 Dec 23 '22 at 03:22
  • The answers depend on what you mean by "reasonable belief." If this translates into a strong prior distribution, such as a Beta$(10, 90)$ distribution, then the Bayesian solution will incorporate that information and give tighter bounds (and a stronger conclusion) than a classical solution. But if the prior is diffuse, such as a Beta$(1/10,9/10)$ distribution, there will scarcely be any difference in the results (although the uncertainties would be expressed differently and be differently interpreted). – whuber Dec 23 '22 at 14:23
  • I believe it would be a strong prior? As in, someone gives you this coin and claims the probability of heads is 1/10. – user1133454 Dec 23 '22 at 15:11
  • That's consistent with both situations I posited. Merely asserting a value for the coin is not necessarily a strong prior. – whuber Dec 26 '22 at 14:58

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