Houjun Liu

Optimal Stopping Problem

  1. Shuffle cards
  2. Keep revealing cards
  3. “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}


\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%”