Card Counting

Behind The Scenes In Over:

0 Simulated Games Of Blackjack

By Tu Nguyen and Eric Banisadr

What is Blackjack & Card Counting?

Preface

Warning

Gambling can be highly addictive and can have severe negative consequences. The following simulations & data do NOT account for real life scenarios and should not be used to model them.

About

With its blinding lights and endless entertainment, Las Vegas attracts millions of tourists yearly. And the main attraction? Casinos. Some lose it all, some make it out with riches. Nevertheless, the house always wins... Or does it?

Black Jack ranks at the top as the most popular game in casinos. While perfect basic strategy still loses long term, clever players have used basic probability to turn the odds into their favor. This is known as card counting.

Game Information

What is Blackjack?

Blackjack is a popular card game featured in casinos where the player is pitted against the house rather than other players. The primary objective of the game is to have a hand value as close to 21 as possible without going over 21.

How do you play?

Card value

The value of each card in the deck is its corresponding number, ie. 2 is a 2, 7 is a 7, etc. Face cards (J, Q, K) have a value of 10, while Aces (A) have a value of 1 or 11.

Terminology

Discard - The area where previously played cards go
Cut - The point at which the dealer will reshuffle the discard into the shoe
Shoe - The card holder with multiple decks in it
Hand - A set of 2 or more cards
Bust - The value of the hand is over 21, considered a loss with no actions remaining

Actions

Stand - The hand value is held and no more actions can be taken on the hand
Hit - Draw a one card from the shoe and add it to total hand value
Double down - When the hand has exactly two cards, up to double your initial bet, then hit once, stand afterwards
Split - When the hand has exactly two cards with the same value (all card with a value of 10 are usually considered pairs with each other), split the pair into two hands with one card from the initial pair in each hand
Surrender - After the initial draw, the player may forfeit the hand, half of the initial bet is returned
Insurance - After the initial draw and if the dealer’s face up card is an Ace (A), the player may place an additional bet up to half of the initial bet that the dealer’s face down card has a value of 10, if correct, double the additional bet is returned

Order of play

Before any cards are dealt, all players must place their initial bet. The dealer then deals cards from the shoe to all players including the dealer. This stops when all players and the dealer have two cards. The dealer will only have one card face up initially. The other card is revealed when all players have finished taking actions. Based on the shown card and the player’s given hand, players will take turns completing actions until they stand or bust on their hand(s). Following this, the dealer will reveal the face down card. If the dealer’s hand value is less than 17, the dealer will hit until they have a hand value of 17 or greater, or they bust.

Shuffling

If the shoe reaches the cut point, the dealer will reshuffle the discard into the shoe and have a player place the cut point (typically between one to two decks from the bottom of the shoe).

How do you win?

Terminology

Push - The player’s hand value is equal to the dealer’s hand value
Blackjack - The initial hand is an Ace (A) and a card with a value of 10
Payout - The return value on the initial bet

Ways to win

♠ The hand has a higher value than the dealer, and it is not a bust
♠ The dealer busts, but the current hand is does not
♠ The initial hand is a Blackjack, and the dealer’s hand was not a Blackjack

If the dealer has the same hand value as the player’s hand is it considered a push and the initial bet is returned to the player. All other cases are considered losses.

Payouts

♣ A winning hand pays out 1:1 on the initial bet
♣ A Blackjack pays 3:2 or 6:5 on the initial bet depending on the casino
♣ Insurance pays 2:1 on the insurance bet

Game Strategy

Basic strategy

Terminology

Edge - percentage advantage one entity has over another
Hard hand - hand with a fixed value
Soft hand - hand with an unfixed value, the hand will contain an Ace (A) that is able to be 1 and 11 without busting

General

Basic strategy outlines the optimal decisions the player should make given the current game state to maximize return on investment. For example, with a hard hand or soft hand, the player would make the corresponding decision based on the hand and the dealer’s face up card in the decision table. However, even with perfect basic strategy, the house still has a roughly 0.5% edge over the player. This is where card counting comes in.

Counting strategy

Terminology

Count value - The value of the card applied to the running count
Running count - The cumulative sum of seen count values
True count - The count determined by the running count divided by the number of remaining decks in the shoe
Hypergeometric distribution - “A discrete probability distribution that describes the probability of k successes in n draws without replacement”
Bankroll - The total amount of money available for the player to bet
Bet spread - The range of possible bets sizes by the player
Balanced system - When all cards are drawn from the shoe, the final running count is 0
Unbalanced system - When all cards are drawn from the shoe, the final running count is not 0

General

Counting strategy combines basic strategy with card counting in order to place optimal bets based on the game state. Since Blackjack draws are based on a hypergeometric distribution, the player is able to gain an edge over the house by keeping track of played cards and betting accordingly. This is done through associating cards with count values, then updating a running count based on the amount of times those cards have been played. From here, a true count is calculated and used to determine the bet size. Counts are reset after a shuffle.

Betting

Bets are determined by a bet spread and the bankroll. The bet spread details the minimum and maximum amount the player can bet. Based on the bankroll, bet spreads are created to maximize the odds of a positive return on investment.

Win Rate Convergence

Strategy

Insights

This animation shows the effect on win rate of each strategy over sessions played. Each session represents 1000 consecutive games of Black Jack. As the amount of sessions increases, we can see the life time win rate across all sessions converge.

Data Generation

Game Settings

General

DAS - Enabled
Max splits - 3
Surrender - Disabled
Insurance - Disabled
Deck penetration - 5

Number of decks - 6
Insurance - Disabled
Hits after ace - 1
Blackjack payout - 3:2
Session games - 1000

Partial Betting

In order to speed up simulations, partial betting was enabled. This is not accurate representation of betting

Dealer Stand on Soft 17

In the simulations, the dealer stands on a soft 17. This is a small advantage for the player.

Simulation Process

Basic Strategy Simulation

For the basic strategy simulations, decisions were basic off tested and proven basic strategy charts.

Counting Strategy Simulation

For the counting strategy simulations, decisions were based off of basic strategy. Bet sizing is based off of true count - 1. Counting systems with associated card count values were provided by Bet and Beat.

Potential Errors & Tradeoffs

♦ Accounting For Multiple Players
♦ Fixed Shuffling
♦ Only Using Balanced Systems

Citations

Websites

Wikipedia - Definitions

Hypergeometric Distribution

Blackjack Apprenticeship

Basic Strategy Decision Charts

Bet and Beat

Counting System Descriptions

Papers

EV of a Card Counter

Jensen, Kamron, "The Expected Value of an Advantage Blackjack player" (2014). All Graduate Plan B and other Reports. 524.

The Probability Behind Blackjack

Blanchard, Hanna, "Blackjack: the math behind the cards" (2019). Louisiana Tech University.