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Put $K=nY$. Apart from a factor independent of $p$, the likelihood isThus the score function and Fisher information matrix are

The Newton method and Fisher scoring updates are respectivelySince $\hat{p}=Y$, initialization at $p^{(0)}=Y$ requires zero steps for either algorithm. Moreover,so binomial-proportion Fisher scoring reaches the MLE in one step from every interior $p^{(0)}\ne Y$.