# Card guessing and the birthday problem for sampling without replacement

@inproceedings{He2021CardGA, title={Card guessing and the birthday problem for sampling without replacement}, author={Jimmy He and Andrea Ottolini}, year={2021} }

Consider a uniformly random deck consisting of cards labelled by numbers from 1 through n, possibly with repeats. A guesser guesses the top card, after which it is revealed and removed and the game continues. What is the expected number of correct guesses under the best and worst strategies? We establish sharp asymptotics for both strategies. For the worst case, this answers a recent question of Diaconis, Graham, He and Spiro, who found the correct order [12]. As part of the proof, we study the…

## One Citation

Online card games

- Mathematics, Computer ScienceElectronic Journal of Probability
- 2022

It is proved that a certain greedy strategy for the shuffler is the unique optimal strategy in this game, and that the guesser can achieve at most log n expected correct guesses asymptotically for fixed m against this greedy strategy.

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