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

2. 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

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

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

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

6. 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

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

8. 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%

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

10. 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)

11. 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

12. 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)

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

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

15. 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.

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

17. 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

18. 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

19. 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

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

21. Questions?

22. 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
Countless other Blackjack sites
Wolfram Alpha, for computing my sums