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If you have a biased coin with probability, p land on the head. You toss 5 times, and it all land on the tail. How to estimate p?

I tried MLE, which gives $p=0$. However, the number of tosses is too small, MLE is not suitable. Is there other way to estimate p?

micatske
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This is covered by the rule of succession. If you assume a uniform prior for $p$, the expected value of the posterior distribution after $s$ heads in $n$ trials is $\frac{s+1}{n+2}$. In your case, with $s=0$ and $n=5$, this is $\frac{0+1}{5+2}=\frac17$. For an elementary derivation of this rule, see How to prove the rule of succession without calculus?.

Note, however, that this estimator is biased (its expected value is $\frac{pn+1}{n+2}=p+\frac{1-2p}{n+2}=p+O\left(\frac1n\right)$), whereas the maximum-likelihood estimator $\frac sn$ is unbiased (its expected value is $p$).

joriki
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    The important assumption here is "assume a uniform prior". – Ethan Bolker Jan 21 '23 at 20:46
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    Ah, that's what it was called! I was trying to find this like a week ago and kept fruitlessly googling relevant terms to try to find it. I know this wasn't the intent of your answer, but thank you for finding the rule of succession for me! – Lieutenant Zipp Jan 21 '23 at 20:48