Probability of Blackjack

Topics: Blackjack, Card game, Playing card Pages: 9 (2179 words) Published: March 12, 2013
Mathematical Studies Project

Probability of Blackjack

Content Page


Statement of task 2

Introduction 3 - 4

Data collection 5 - 6

The four Blackjack strategies 7 - 15

Conclusion 16

Bibliography 17

Statement of Task

This project aims to investigate how mathematics works in one of the most popular card games in the world - Blackjack. It examines the probability in the Blackjack and how the use of mathematics can be applied into the game in order to increase the winning probability of the player and lower the house edge of the casino. Currently, there are more than a hundred rules and strategies being used by Blackjack players in casinos. This project will focus on the four most commonly used strategies of Blackjack including the dealer’s upcard strategy, the Hit or Stand strategy, doubling down and splitting, and the Mimicking Strategy.

The project will firstly give a brief introduction of the game Blackjack. It will then outline the basic rules and define the basic terminology of Blackjack. After that, it will look into each of the four strategies and their impact on house edge and player’s edge. The probability of busting and the probability of Blackjack will also be examined in the project. The project will end with a conclusion on the sources of error within the project and how might the four strategies not work in today’s multi-deck Blackjack. The ultimate goal of the project is to lower the casino edge and increase the player’s advantage and thus the chances of winning of the player. This project about Blackjack will show readers how mathematics can be applied into our everyday lives.

Introduction of Blackjack

As mentioned before Blackjack is currently the most popular card game in the world. Much of the popularity is gained by the fact that it is simple, straightforward yet beatable. By playing the game with the basic strategy, the player is able to minimize the house advantage and increase the odds of winning. This is why the game is also considered as “defeatable” and profitable.


The basic premise of the game is that you want to have a hand value that is closer to 21 than that of the dealer, without going over 21. If a value of less than 21 is dealt, the player may choose to be dealt single cards until they either reach a value of 21(Blackjack), reach a value they feel comfortable to play, or reach a value that exceeds twenty-one. The winner holds a hand with a value of, or nearest to, twenty-one without exceeding it. The dealer must hit until he gets a hand value of 17 or more or busts by going over 21. The rules of play for the dealer are strictly dictated, leaving no decisions up to the dealer. The cards in Blackjack are valued as follows: • An ace can count as either one or eleven.

• 2 to 9 are valued at their face value.
• 10, Jack, Queen and King are all valued at ten.

Basic knowledge of Blackjack

Let x be the player’s sum of card values of the first two cards, where [pic].

Let D be the dealer’s upcard value, where [pic] .

Let T be the final total obtained by the dealer and J be the total obtained by the player, where [pic].


Case 1
If T > 21 or T < J, the player who stands on J wins.

Case 2
If J < T < 21 the player loses

Case 3
If J = T it is considered a tie

Blackjack glossary

Hit: The player takes another card from the dealer.

Stand: Stop and not to be dealt any more cards. The hand that stands is the one that will be compared to the...
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