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Title: Texas Hold 'Em Strategy
Description: This is a math paper on the strategy of texas hold em for all poker players to understand
Description: This is a math paper on the strategy of texas hold em for all poker players to understand
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Ryan Passaro
Math Fair 2016
Commack Middle School
Abstract
In my project, I will explain the basic terms and rules of Texas Hold‘em
...
There decisions will be based
upon principles of probability, combinatorics, expected value and pot odds
...
In 1967, after the card game
spread throughout Texas, it was introduced to Las Vegas by a group of people
...
Addington
said he first saw the game in 1955
...
Van De Kamp
...
Crandell Addington is an entrepreneur and a famous poker player, known for
being one of the founders of the WSoP (World Series of Poker) and a member of the
Poker Hall of Fame
...
He has won the WSOP twice and was an inductee in the Poker Hall of Fame
...
In
1979, he wrote his first book on poker strategy and wrote seven more, one of his
last ones on online poker
...
He was
a professional poker player who was inducted into the Hall of Fame in 1992
...
Objective
Cards are dealt randomly a total of two cards to each player and five
community cards
...
The objective of Texas
Hold ‘em is to win as many hands as a player can win by mathematically making
better decisions
...
Each player can attempt to control the amount of money in the pot
better if they can use mathematics in their strategy
...
Bluffing is a technique not previously mentioned and often used in Texas Hold ‘em
...
An example would
be to raise even if you don’t have a good hand to try and convince the other players
your hand is better than it actually is
...
Television and internet
were two reasons why at the turn of the century Texas Hold ‘em’s popularity
skyrocketed
...
Texas Hold ‘em was also in many
movies such as; Rounders (1998), Casino Royale (2006), and Shark out of Water
(2008), just to name a few
...
This debate is greater explained towards
the end of my introduction
...
These are
bets that two people must put in during the pre-‐flop
...
In some cases, the game is also played with antes
...
The difference of this and the blind bets is that the big/small
blinds must put in their bet before folding
...
After one round of betting is done, the next betting round will
start by the person in the small blind
...
The big blinds’ bet is equivalent to the other player’s antes
...
During the “head to head” time,
the player in the dealer position is the small blind
...
Play of the Hand
The game begins with two cards dealt face-‐down to each player
...
This time is called the pre-‐flop
...
Next, the first the community cards are dealt
...
Any player may use these cards
...
Another betting round
then occurs
...
The turn is when one
more community card is dealt
...
Finally the last
community card is dealt called the “river” or “fifth street” with the last betting
round
...
Every time a new community card is
dealt, a card is ‘burned’
...
Strategy
Most poker professionals recommend an aggressive approach to a game
...
Mostly all poker authors agree that your style of
play depends on where your position is around the table
...
In conclusion, people who have earlier positions usually stay in less rounds
...
Famous
Hold ‘em player Phil Hellmuth once said, “If it weren’t for luck, I’d win every time
...
On the flip
side, many other people believe that no matter how much skill is involved you
cannot predict what will happen next, nor control the future
...
It can
be calculated by:
𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 =
# 𝑜𝑓 𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
# 𝑜𝑓 𝑡𝑜𝑡𝑎𝑙 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
For example, to find the probability of picking a prime number out of a group
of numbers from 1-‐12, this would be the equation:
#"of"favorable"outcomes:"in"this"case,"
any"prime"number"between"1712"
(2,"3,"5,"7,"11)"
#"of"total"possible"outcomes:"in"
this"case,"any"number"from"1712"
So the probability of picking a prime number out of the numbers 1-‐12 inclusive
would be 5 out of 12, or 41
...
2
5
Expected Value
What is expected value? Expected value is the predicted value of a variable
...
Let’s plug in
that information into our equation;
𝐸𝑉 = % 𝑜𝑓 𝑤𝑖𝑛𝑛𝑖𝑛𝑔 10 + (% 𝑜𝑓 𝑙𝑜𝑠𝑖𝑛𝑔)(3)
Now, as you can see, we need to calculate the percentage of losing and the
percentage of losing;
Probability*
Probability*
Probability*
Probability*
of*rolling*
of*rolling*
of*rolling*
of*rolling*
Probability*
double*
double*
double*
double*sixes*
of*rolling*
ones*
twos*
fours*
doubles*
Probability*
Probability*
of*rolling*
of*rolling*
double*
double*fives*
threes*
!
So the odds of rolling doubles are !
...
Let’s plug that in
...
6
6
6
6
6
If the expected value is positive you should play because even though you can lose
𝐸𝑉 =
sometimes, you will make money in the long run
...
So in this case, you should not play
...
In a jar, there are 11 blue dice and 16 red dice
...
Basic Permutation and Combination Skills
What is a permutation? A permutation is a way that a set of numbers can be
arranged when order matters
...
It means every number from itself to 1 is multiplied
...
Example:
How many different ways can 5 books, (Math, English, Spanish,
Chemistry, and Physics) be arranged on a shelf in groups of 4?
Total&number&of&books&
This can be written as;
!!
!! !∙!∙!∙!∙!
= =
(!!!)! !!
!
Number&of&books&in&a&group&
=5 ∙ 4 ∙ 3 ∙ 2=120
So there are 120 ways to arrange four books on a shelf
...
What is a combination? A combination is a way that a set of numbers can be
! 𝑃
!
=
arranged when order doesn’t matter
...
Calculating A Hand
Let’s calculate the chance of getting a Full House:
A full house is a three of a kind and a pair within your five cards
...
So we have 13 choices and we are choosing one of them, because
we need a card number for the first three cards
...
We also have four
suits and we are choosing 3 or have 3 choices for
...
We
multiply these numbers together
...
We are choosing
between 12 different card numbers now because we cannot have a four of a kind
...
We also have 4 suits and are choosing 2
...
Now, we multiply them all;
!" 𝐶!
∙
! 𝐶!
∙ !" 𝐶! ∙
! 𝐶!
13!
4!
12!
4!
∙
∙
∙
1! 13 − 1 ! 3! 4 − 3 ! 1! 12 − 11 ! 2! 4 − 2 !
13! 4! 12! 4!
∙ ∙
∙
12! 3! 11! 4
13 ∙ 4 ∙ 12 ∙ 6 = 3,744
Next, we have to divide this number by the total possible hands you could
have
...
2,580,960 is calculated by there are 52 cards in a
deck, and 5 cards in a hand
...
!" 𝐶!
= 2,598,960
3,744
1
= 694
2,598,960
6
!
So the odds of getting a full house are 1 in 694!
...
This is the skill
portion of the game
...
Should I call, raise, or fold?
The only possible hand that your opponent could get that would beat you would be
the full house
...
My%2%cards%+%
opponents%2%
cards%+%4%flop%
There%is%a%
cards%are%
King%of%
known
...
Should you call? You
should only call if your odds of winning are greater than your pot odds
...
Your pot odds are your contribution to the pot to play as a fraction of the
total
...
Raising has two purposes
...
!
If I put myself in my opponents shoes, my odds of winning is !!, what raise
would I have to call, to make my pot odds greater than my implied odds? (Only then
would I play)
...
11𝑥 > 220 + 2𝑥
220 + 2𝑥 11
9𝑥 > 220
4
𝑥 > 24
9
So if you bet anything above $25, he should fold
...
Conclusion
I am very interested in this topic and that is why I chose to study the
mathematics behind the game of Texas Hold’em; my family and I play often, so I
hope to use the skills I’ve learned in the future
...
Expected value calculations can help me decide when
hands should be played or folded
...
Also, I realized how math is used in our everyday lives: lottery, insurance
policies, or any risk analysis involves similar expected value calculations
...
If I had more time to continue my research I would have situations with
more than two people, situations on the turn or different poker card games
...
com
...
” Poker Terms and Definitions
...
p
...
Web
...
2015
...
pokerstars
...
I used this website to explains terms of Texas Hold’em
...
“Permutations and Combinations
...
N
...
, n
...
Web
...
2015
...
regentsprep
...
htm
...
Young, Wesley R
...
” HowStuffWorks
...
com, 07 Apr
...
Web
...
2015
...
hotstuffworks
...
htm
...
Cotter, Neil E
...
” (n
...
): n
...
1 Feb
...
Web
...
2015
...
ecu
...
edu/eceCTools/Probability/Combinatorics/ProbCombEx15
...
I used this website to explain calculating the odds of getting a hand of poker
...
Doyle Brunson's Super System: A Course in
Power Poker
...
, 2002
...
I used this book to get a better understanding of poker strategy
Title: Texas Hold 'Em Strategy
Description: This is a math paper on the strategy of texas hold em for all poker players to understand
Description: This is a math paper on the strategy of texas hold em for all poker players to understand