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@jemoka / Jemoka Knowledge Base / wiki/concepts/optimal_stopping_problem.md
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--- title: "Optimal Stopping Problem" type: concept related: [Martingale Model] source: https://www.jemoka.com/posts/kbhoptimal_stopping_problem/ confidence: high status: active --- Shuffle cards Keep revealing cards “Stop” when there’s >50% chance the next card to be revealed is black We can Frequentist Definition of Probability calculate the probability of a given card remaining is black: \begin{equation} pblack(b,r) = \frac{26-b}{52-(r+b)} \end{equation} now: \begin{equation} pwin(b,r) = \begin{cases} 0, b+r = 52 \\ \max \qty[ \begin{align}&pblack(p,r), \\ &pblack(b,r)pwin(b+1,r) + (1-pblack(b,r)pwin(b, r+1) \end{align}] \end{cases} \end{equation} “with the theory of the Martingales, this comes out to be 50%”