On Card guessing after a single shelf shuffle

TL;DR

This paper analyzes the distribution of correct guesses in a card game after a single shelf shuffle, providing new distributional results, a CLT, and extensions to biased shuffles, enhancing understanding of shuffle randomness.

Contribution

It offers a complete distributional analysis of correct guesses after one shelf shuffle, including a CLT and extensions to biased shuffles, with detailed position matrices.

Findings

Expected number of correct guesses re-derived

Distribution of correct guesses fully characterized

Central limit theorem established for large n

Abstract

We consider a card guessing game with complete feedback. An ordered deck of $n$ cards labeled $1$ up to $n$ is shelf-shuffled exactly one time. One after the other a single card is drawn from the shuffled deck. The guesser makes has guess and the card is shown until no cards remain. We provide a distributional analysis of the number of correct guesses under the optimal strategy. We re-obtain the previously derived expectation and add a complete description of the distribution. We also obtain a central limit theorem for the number $n$ of cards tending to infinity. Furthermore, we discuss an unbalanced, biased shelf shuffle and show how to derive the extend our analysis, also adding the complete position matrix. Finally, a refined analysis of the number of correct guesses is carried out, distinguishing between pure luck guesses and certified correct guesses.