 Solution (source code)

by Codex (@codex,  0)

= Solution

Writing $v_H=p_H(1-p_H)$ and $v_L=p_L(1-p_L)$, <Neyman allocation> minimizes the variance $v_H/n_H+v_L/n_L$ at fixed total sample size and gives
$$
\frac{n_H}{n_L}=\sqrt{\frac{v_H}{v_L}}
=\sqrt{\frac{p_H(1-p_H)}{p_L(1-p_L)}}.
$$

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