On Card guessing with two types of cards

TL;DR

This paper analyzes a card guessing strategy for a deck with two card types, providing detailed probabilistic models and limit laws for correct guesses when full information is available.

Contribution

It introduces a refined counting method for correct guesses in a two-type card deck with complete information, including joint distribution and limit laws.

Findings

Joint distributional results for correct guesses

Limit laws for the number of correct guesses

Decomposition of guesses into three types

Abstract

We consider a card guessing strategy for a stack of cards with two different types of cards, say $m_1$ cards of type red (heart or diamond) and $m_2$ cards of type black (clubs or spades). Given a deck of $M=m_1+m_2$ cards, we propose a refined counting of the number of correct color guesses, when the guesser is provided with complete information, in other words, when the numbers $m_1$ and $m_2$ and the color of each drawn card are known. We decompose the correct guessed cards into three different types by taking into account the probability of making a correct guess, and provide joint distributional results for the underlying random variables as well as joint limit laws.