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Dealing until kings or a king-and-ace

HardCombinatoricsExpected Value~12m

Cards are dealt one at a time from a shuffled standard 5252-card deck. Dealing stops as soon as either two kings have appeared, or at least one king and at least one ace have appeared - whichever comes first. Find the expected number of cards dealt.

Show hints (2)+
  1. E[cards]=∑m≥0P[still going after m]\mathbb{E}[\text{cards}]=\sum_{m\ge0}\mathbb{P}[\text{still going after }m]. You survive iff 00 kings (any aces) or 11 king and 00 aces.
  2. Survival =[(48m)+4(44m−1)]/(52m)=\big[\binom{48}{m}+4\binom{44}{m-1}\big]/\binom{52}{m}; the sum is 121990≈13.54\tfrac{1219}{90}\approx13.54.

Answer

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13.5444 (± 0.05)

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Asked at: Jane Street, Citadel

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