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We want to find the convex combination $S$ of iid Bernoulli random variablesthat maximizes $\textbf{P}(S\geq t)$ for a given threshold~$t$. Cs\'okaconjectured that such an $S$ is an average if $t\geq p$, where $p$ is thesuccess probability of the Bernoulli random variables. We prove this conjecturefor a range of $p$ and $t$.

Optimizing stakes in simultaneous bets