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Suppose you have a bag of 100 coins of which 1 is biased with both sides as Heads. You pick a coin from the bag and toss it three times. The result of all three tosses is Heads.

What is the likelihood that the selected coin is biased? What is the MLE?

What I think but not sure:

Likelihood that the selected coin is biased: p(HHH|biased) = 1 * 1 * 1 = 1

MLE: max { p(HHH|biased), p(HHH|fair) } = max { 1, 1/8 } = 1

Should I take into account the fact that there is only 1/100 biased?

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    $\begingroup$ The 'likelihood' part is pretty basic. Can you do that part? // For 'textbook style' problems, we expect you to show what you have tried. Please 'take the tour' and consider adding the Self-study tag. // If you toss a coin three times you don't really have enough evidence to give a good guess whether it's fair or not. However, if you get Heads three times in a row from any coin, do you begin to suspect it might not be fair? $\endgroup$ – BruceET Sep 28 '18 at 23:37
  • $\begingroup$ Yes but there is only 1/100 biased coin. Would this fact affect the likelihood? $\endgroup$ – David Sep 29 '18 at 1:08
  • $\begingroup$ If you look at the "Related" items in the right margin of the page (beneath the ads), you will see various treatments (some Bayesian) of similar problems. Your problem seems imprecisely stated, perhaps on purpose to make you think of various approaches. // For example, "What is the MLE?" (Of what?) If that's the MLE of heads probability $\theta,$ then likelihood fcn for 3H's is max at $\hat \theta = 3/3 = 1,$ which takes into account only the one coin observed. // "What is likelihood coin unbiased?" Does that mean "What is probability coin unbiased?" Then use discrete version of Bayes' Thm. $\endgroup$ – BruceET Sep 29 '18 at 8:20
  • $\begingroup$ How do you define a MLE in a problem with no parameter? $\endgroup$ – Xi'an Oct 1 '18 at 11:17
  • $\begingroup$ I think the parameter in this question is if the coin is biased or not. $\endgroup$ – David Oct 3 '18 at 11:18

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