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Guessing cards from a known pile

HardProbabilityExpected Value~10m

A pile holds 33 aces and 33 jacks. Cards are drawn one at a time without replacement; before each draw you guess its identity, earning $1 per correct guess. Playing optimally over all 66 draws, what is your expected total earnings?

Show hints (2)+
  1. Optimal play: guess whichever type has more cards left. Set up E(a,j)E(a,j).
  2. E(a,j)=max⁡(a,j)a+j+aa+jE(a−1,j)+ja+jE(a,j−1)E(a,j)=\tfrac{\max(a,j)}{a+j}+\tfrac{a}{a+j}E(a-1,j)+\tfrac{j}{a+j}E(a,j-1); E(3,3)=4.1E(3,3)=4.1.

Answer

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4.1 (± 0.02)

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Asked at: Jane Street, Citadel

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