No-feedback Card Guessing Game: Moments and distributions under the optimal strategy

TL;DR

This paper derives exact formulas for the expected correct guesses and moments in a no-feedback card guessing game under optimal strategy, addressing open problems for multiple shuffles.

Contribution

It provides a closed-form formula for the 1-shuffle case and a combinatorial proof for multiple shuffles, solving an open problem in the field.

Findings

Exact expected number of correct guesses for 1-shuffle case

Combinatorial proof for multiple shuffles

Solution to an open problem in card guessing game analysis

Abstract

Relying on the optimal guessing strategy recently found for a no-feedback card guessing game with $k$-time riffle shuffles, we derive an exact, closed-form formula for the expected number of correct guesses and higher moments for a $1$-time shuffle case. Our approach makes use of the fast generating function based on a recurrence relation, the method of overlapping stages, and interpolation. As for $k>1$-time shuffles, we establish the expected number of correct guesses through a self-contained combinatorial proof. The proof turns out to be the answer to an open problem listed in Krityakierne and Thanatipanonda (2022), asking for a combinatorial interpretation of a generating function object introduced therein.