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Is there a way to determine whether a coin is biased, using probability/statistics method, say the following two questions:

if observe 8 heads in 10 flips, is the coin biased? Or

if observed 3 sequences of 5 flips. At least one sequence was all heads. Is the coin biased?

Are there some general methods to deal with these kinds of problems?

Nick Cox
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william007
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1 Answers1

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P-value = prob. of observe 8 heads in 10 flips + prob. of observe 9 heads in 10 flips + prob. of observe 10 heads in 10 flips = 0.043 + 0.010 + 0.001 = 0.054.

Prob. of all heads in 5 flips 0.03125

p-value = prob of At least one sequence was all heads = 1 - $(1-0.03125)^3$ = 0.09

The meaning of the p-value is: Assume the coin is fair (50% chance of head), the probability of getting observed output or more extreme output.

For the first case, if the coin is fair, the chance that you get 8 or more heads among 10 flips is 5.4%. Do you believe that coin is fair? Make your judgement by yourself.

The general principle is following binomial distribution for flipping coin thing.

user158565
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  • The application of a one-sided test to this question is invalid. – whuber Nov 28 '18 at 16:24
  • I think using one-side thing giving you already known the results, and using two-side thing before you know the results. – user158565 Nov 28 '18 at 16:54
  • That's post-hoc hypothesizing--which is why the p-values you get are invalid. – whuber Nov 28 '18 at 17:29
  • Could we associate some kinds of statistical test (like t-test, here seems like it is based on binomial distribution, do we have a test here on binomial distribution) on this experiment to accept/reject the hypothesis? Say if unbiased coin has a P=0.5 for heads (or tails), a hypothesis test could be set up such that H0: Coin is fair; and H1: Coin is not fair. – william007 Nov 29 '18 at 01:35
  • The p-values I gave in the Answer is the p-value to test the null hypothesis that coin is fare. (one sided). If you set alpha at 0.05 level, On bother situations, you cannot reject that coin is fair. t-test or z-test or Chi-square test is not good. All of them are some kinds of approximation based on large sample. In your situation, he sample size is not large enough to used them. – user158565 Nov 29 '18 at 02:20