When looking at questions on X validated, I came across this seemingly obvious request for an unbiased estimator of P(X=k), when X~B(n,p). Except that X is not observed but only Y~B(s,p) with s<n. Since P(X=k) is a polynomial in p, I was expecting such an unbiased estimator to exist. But it does not, for the reasons that Y only takes s+1 values and that any function of Y, including the MLE of P(X=k), has an expectation involving monomials in p of power s at most. It is actually straightforward to establish properly that the unbiased estimator does not exist. But this remains an interesting additional example of the rarity of the existence of unbiased estimators, to be saved until a future mathematical statistics exam!