Guessing cards from a known pile

HardProbability Expected Value~10m

A pile holds $3$ aces and $3$ 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 $6$ draws, what is your expected total earnings?

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Optimal play: guess whichever type has more cards left. Set up $E(a,j)$ .
$E(a,j)=\tfrac{\max(a,j)}{a+j}+\tfrac{a}{a+j}E(a-1,j)+\tfrac{j}{a+j}E(a,j-1)$ ; $E(3,3)=4.1$ .

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4.1 (± 0.02)

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Interview format: Market Maker, Hedge Fund Quant

Guessing cards from a known pile

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