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The Mathematics Of Bluffing In Poker

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Words 1900
Pages 8
The Mathematics of Bluffing in Poker

Wataru Hasegawa
000040-0111
United World College of South East Asia
SL Math Exploration
Stephen Lumb
Contents
Introduction …3

Rationale …3

Aim …3

Basics …3~4

Unexploitable Bluffing …4~5

Numerical Calculations …5~6

Graphical and Calculus Calculations …6~7

Conclusion …8

Evaluation …8

Works Cited …9

Introduction
Many card games can improve the chances of winning through mathematical manipulation and the understanding of probabilities. This is because throughout most games you are required to pull the right card in order …show more content…
Not only did it intrigue me because I did not believe the hypothesis but because I played every once in a while and thought that exploring this topic would widen my knowledge of the game resulting in the slightest improvement.

Aim
This exploration aims to combine a wide variety of ideas and methods such as probability and calculus to determine an underlying mathematical strategy for good bluffing in poker.

Basics
In order to understand the mathematics behind bluffing, first you need to understand how the game works. However poker is not as simple as you may think and requires more than just an explanation to fully comprehend what is going on. The simplest possible version of poker requires the use of only 3 cards and 2 players, whereas the full game would require all 52 cards and an average of 5 players. For example, imagine three cards, King, Queen and Jack, all the same suit. Now you and a friend will place $1 bet to start the game. From the 3 shuffled cards, 1 random card will be dealt to each player, leaving a single unknown card remaining. After both players have acknowledged their own card, one player will proceed to take the second …show more content…
However, if either player gets the Queen, things get complicated because the player does not know whether their opponent has the card with a higher or lower value. This means you or your friend can start bluffing creating a third option varying certain components of the game.

If your friend possesses the Queen he could place a bet to bluff you into thinking he has the King, when in reality he may have the Jack. However there is a flaw if one player continuously gets to start as the second player will always be at a disadvantage, so in order to create an equal stand point, both players will have to alternate roles.

Unexploitable bluffing Figure 1 – A tree diagram showing all possible outcomes in a poker game consisting of 3 cards and 2 players (Ace, King, Queen)

Through the use of game theory and probability Mathematician John Nash proved that there exists a mathematical bluffing technique he called “unexploitable mixed strategy”. He called it unexploitable because neither player could improve their win rate.

In the diagram the first branch represents the different cards the second player can draw, each possibility being .

The second branch represents the card the first player can have, each possibility being

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