Blackjack: Counting Hands and Counting Cards Presentation by Joe Brown, credit to Edward O. Thorp and Catalin Barboianu

Outline How to actually play blackjack Difficulty in modelling games How to play I: The actual game How to play II: Helpful terms How to play III: Variations of importance Difficulty in modelling games A simple example (Odds of getting a natural) A more complex example (possible game starts) What makes this so hard Counting cards Counting is a misnomer Let’s learn how to count Running vs True

Playing and Winning at Blackjack What is your win condition? How do you achieve your win condition?

A bit of lingo to help us out “Natural/blackjack” “Soft” vs “Hard” Shoe Game Options Hit Stay Double (Down) Split Surrender Insurance

Variations on the game How many decks are being used? When does dealer hit? Do we have hole cards? Resplitting and Reno rule

Dealing and how it affects counting Dealing individual hands gives us combinations Dealing cards “in order” gives us permutations For now (this presentation), we’ll make our lives simpler

Simple example (premise) What are the odds of my opening hand being natural with one deck?

Simple example (math) First take all possible first hands C(52,2)=1326 Now, pick our ace C(4,1)=4 Now pick our “10” C(16,1)=16 Final division (C(16,1)*C(4,1))/C(52,2)=64/1326≈5%

One to work through together What are the odds of my initial hand being able to be split?

Complex example (premise) Now that we know the odds of an initial natural on one deck, let’s calculate the number of possible opening hands from any number of players from any number of decks up to 8. Caveat: To make life easier, we assume all cards are distinct (e.g. 10 of clubs from deck 1 is separate from 10 of clubs from deck 3)

Complex example (math) Let’s try one deck at first. So we take the possible first hand for k players, and then sum up the k’s

Increasing the difficulty So one deck wasn’t too hard to calculate, now lets try two. Same way as before, just double some numbers (cards and players)

The next logical summation Well, if we can calculate each individual deck number, why not sum them?

The Grand Total The number of possible opening hands of blackjack (using anywhere from 1 to 8 decks) is 2019207372 This is about 3/10 the amount of people living today, so not incomprehensibly large, but still big

Why we chose combinations Not only does it make hand order not matter like in true blackjack, but it means that the numbers are smaller. For example, if we did permutations for 8 decks and the maximum number of players, we’d have (52*8)!≈3.8*10910 hands.

The crux of the modeling problem Sheer size You can’t keep hitting forever Just the sheer number of options

Learning how to “cheat” by addition Counting cards isn’t keeping track of every card, it’s just adding Assign every card a number 2-6=+1 7-9=0 10-Ace=-1

Running Count vs True Count Just because you have a +5 on board, this doesn’t mean your odds are that much greater if you have 7 decks in the shoe. Thus, we divide

Why counting works It increases chance of natural It increases the odds of splitting being worth it It makes doubling likely more worth it It makes the dealer more likely to bust It makes insurance worth more. However, counting isn’t perfect, so one still needs to be smart

Improvements and variations in general Other counting systems More in depth combinatorics Computer programs

Questions?

References Beat the Dealer: A Winning Strategy for the Game of Twenty-One, Edward O. Thorp Probability Guide to Gambling: The Mathematics of Dice, Slots, Roulette, Baccarat, Blackjack, Poker, Lottery and Sport Bets, Catalin Barboianu https://www.blackjackapprenticeship.com/how-to-count-cards/ https://en.wikipedia.org/wiki/Card_counting https://en.wikipedia.org/wiki/Blackjack Countless other Blackjack sites Wolfram Alpha, for computing my sums