Dealing until kings or a king-and-ace

HardCombinatorics Expected Value~12m

Cards are dealt one at a time from a shuffled standard $52$ -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.

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

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13.5444 (± 0.05)

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Dealing until kings or a king-and-ace

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