On Card guessing games: limit law for one-time riffle shuffle

TL;DR

This paper analyzes a card guessing game after a single riffle shuffle, establishing a limit law for correct guesses and connecting it to a two-color guessing game, revealing distribution properties of luck-based guesses.

Contribution

It provides a new limit law for the number of correct guesses after one riffle shuffle and links it to a two-color guessing game, improving previous results.

Findings

Limit law for correct guesses established

Connection made to two-color guessing game

Distribution result for luck-based guesses derived

Abstract

We consider a card guessing game with complete feedback. A ordered deck of n cards labeled 1 up to n is riffle-shuffled exactly one time. Then, the goal of the game is to maximize the number of correct guesses of the cards, where one after another a single card is drawn from the top, and shown to the guesser until no cards remain. Improving earlier results, we provide a limit law for the number of correct guesses. As a byproduct, we relate the number of correct guesses in this card guessing game to the number of correct guesses under a two-color card guessing game with complete feedback. Using this connection to two-color card guessing, we can also show a limiting distribution result for the first occurrence of a pure luck guess.