10 centuries of the history of the poker. Basic rules and mathematical expectation. What should you do to win more often. For information on other games visit https://nz-casinos.com/.

1. Mathematics and theMathematics and the
Game of PokerGame of Poker
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2. The History of PokerThe History of Poker
 Over the past 10 centuriesOver the past 10 centuries
poker has evolved frompoker has evolved from
various gamesvarious games
– 969 AD: Emperor Mutsung in969 AD: Emperor Mutsung in
ChinaChina
– 1212thth
& 13& 13thth
centuries: Eyptianscenturies: Eyptians
– 1616thth
century: “Primero” is oftencentury: “Primero” is often
called “poker’s mother”called “poker’s mother”
 Each player was dealt 3 cards andEach player was dealt 3 cards and
bluffing was a very large part ofbluffing was a very large part of
the gamethe game

3. The History of PokerThe History of Poker
 In the U.S.In the U.S.
– 1834: Being played on1834: Being played on
Mississippi RiverboatsMississippi Riverboats
 Referred to as theReferred to as the
“cheating game”“cheating game”
― Civil War: extremely popular with soldiersCivil War: extremely popular with soldiers
for both the North and Southfor both the North and South
― Wild West period: poker table found in aWild West period: poker table found in a
saloon in almost every town across thesaloon in almost every town across the
countrycountry

4. The Different Games ofThe Different Games of
PokerPoker
1)1) 5 Card Draw – grew in popularity after the5 Card Draw – grew in popularity after the
Civil War and remained the most popularCivil War and remained the most popular
for almost a centuryfor almost a century
2)2) 7 Card Stud – shorty before WWII became7 Card Stud – shorty before WWII became
the most popular and remained so for 40the most popular and remained so for 40
yearsyears
3)3) Texas Hold ‘Em – became the dominantTexas Hold ‘Em – became the dominant
game in the 1970’s. Most prominent gamegame in the 1970’s. Most prominent game
of poker in the world.of poker in the world.
-hundreds of forms of poker exist-hundreds of forms of poker exist

5. Basic Rules ofBasic Rules of
Texas Hold ‘EmTexas Hold ‘Em
 The point of poker is to make moneyThe point of poker is to make money
– ““when the cards are dealt; you are nowhen the cards are dealt; you are no
longer a grandson, a friend, or a nicelonger a grandson, a friend, or a nice
guy; you are a player” (Sklyansky)guy; you are a player” (Sklyansky)
1)1) Post big blind and little blindPost big blind and little blind
2)2) Dealer deals each player 2 cards face downDealer deals each player 2 cards face down
3)3) Betting begins – can call, raise, or foldBetting begins – can call, raise, or fold
4)4) The FlopThe Flop – the dealer burns the top card and– the dealer burns the top card and
places 3 cards on table face up. 2places 3 cards on table face up. 2ndnd
round of bettinground of betting

6. Basic Rules ofBasic Rules of
Texas Hold ‘EmTexas Hold ‘Em
5)5) The TurnThe Turn – burns a card and another card– burns a card and another card
placed face up on table. 3placed face up on table. 3rdrd
round of bettinground of betting
6)6) The RiverThe River – burns a card and places the– burns a card and places the
last card face up on table. 4last card face up on table. 4thth
and finaland final
round of bettinground of betting
7)7) A player can use any combination of the 7A player can use any combination of the 7
available cards – 5 community cards and 2available cards – 5 community cards and 2
in hand – to make best 5 card poker handin hand – to make best 5 card poker hand
8)8) Hands are revealed. The best hand wins.Hands are revealed. The best hand wins.

8. Mathematical ExpectationMathematical Expectation
 Known as the expected value inKnown as the expected value in
Statistics, though name is misleadingStatistics, though name is misleading
 Generally not a value that will beGenerally not a value that will be
achievedachieved
 Better to think of it as the long termBetter to think of it as the long term
average value of the variable overaverage value of the variable over
numerous independent trialsnumerous independent trials
 In poker: the amount a bet willIn poker: the amount a bet will
average winning or losingaverage winning or losing

9. Mathematical ExpectationMathematical Expectation
 Example:Example:
– betting a friend $1 on the flip of a coin. Each timebetting a friend $1 on the flip of a coin. Each time
it comes up head, you win. Each time it comes upit comes up head, you win. Each time it comes up
tails, you lose.tails, you lose.
 The odds of coming up heads are 1-to-1The odds of coming up heads are 1-to-1
 You are betting $1-to-$1You are betting $1-to-$1
 Mathematical Expectation = 0Mathematical Expectation = 0
– Cannot expect to be ahead or behind after 2 flipsCannot expect to be ahead or behind after 2 flips
or 200 flipsor 200 flips
 Expectation = (w * pw) + (-v * pl)Expectation = (w * pw) + (-v * pl)
– w = gain on the winning betw = gain on the winning bet
– pw = probability of the winpw = probability of the win
– v = value of the lossv = value of the loss
– pl = probability of the losspl = probability of the loss

10. Mathematical ExpectationMathematical Expectation
 Now, say your friend (who is not tooNow, say your friend (who is not too
intelligent) wants to bet $2 to your $1 on theintelligent) wants to bet $2 to your $1 on the
flip of a coinflip of a coin
 Do you take the bet?Do you take the bet?
 The odds of coming up heads are still 1-to-1The odds of coming up heads are still 1-to-1
 You are now betting $2-to-$1You are now betting $2-to-$1
 Mathematical Expectation = $0.50Mathematical Expectation = $0.50
– Expect to win one and lose oneExpect to win one and lose one
– Lose first one, lose $1Lose first one, lose $1
– Win second one, win $2Win second one, win $2
 By the equation:By the equation:
– E = (2 * ½) + (-1 * ½) = ½ = $0.50E = (2 * ½) + (-1 * ½) = ½ = $0.50

11. Mathematical ExpectationMathematical Expectation
 A person chooses a number between 1A person chooses a number between 1
and 5 and holds it behind their back.and 5 and holds it behind their back.
They bet you $5 to your $1 that youThey bet you $5 to your $1 that you
cannot guess the number.cannot guess the number.
 Do you take the bet?Do you take the bet?
 What is the mathematical expectation?What is the mathematical expectation?

12. Mathematical ExpectationMathematical Expectation
 w = $5w = $5
 pw = 1/5pw = 1/5
 v = $1v = $1
 pl = 4/5pl = 4/5
 E = (5 *E = (5 * 11
//55)+(-1 *)+(-1 * 44
//55) = 1/5 =$0.20) = 1/5 =$0.20

13. Mathematical ExpectationMathematical Expectation
 In poker, it allows players to predictIn poker, it allows players to predict
how much money they are going tohow much money they are going to
win, or losewin, or lose
 The calculation of mathematicalThe calculation of mathematical
expectation, money managementexpectation, money management
skills, and knowing the outs and potskills, and knowing the outs and pot
odds allows a player to play aodds allows a player to play a
profitable gameprofitable game

14. Pot Odds & OutsPot Odds & Outs
 Outs: the number of cards left in theOuts: the number of cards left in the
deck that will improve your handdeck that will improve your hand
– Ex: you have 4 spades on the Turn, so youEx: you have 4 spades on the Turn, so you
have 9 outs left to get the flush on thehave 9 outs left to get the flush on the
RiverRiver
 Pot odds: the ratio of the amount ofPot odds: the ratio of the amount of
money in the pot to the bet you mustmoney in the pot to the bet you must
call to continue in the handcall to continue in the hand
– Ex: If there is currently $1000 in the potEx: If there is currently $1000 in the pot
and you have to put in $20 to call, yourand you have to put in $20 to call, your
pot odds are 1000:20 or 50:1pot odds are 1000:20 or 50:1

15. Odds with Exposed &Odds with Exposed &
Unseen CardsUnseen Cards
 When figuring the outs, why are theWhen figuring the outs, why are the
burned cards and the number of cardsburned cards and the number of cards
your opponents have not considered?your opponents have not considered?
– Consider all unseen cards as potential outs!Consider all unseen cards as potential outs!
 Say you have 2 cards and your friend has 10Say you have 2 cards and your friend has 10
 You get to draw 1 more card from the remainingYou get to draw 1 more card from the remaining
deck of 40 cardsdeck of 40 cards
 The odds of that 1 card being the Ace of ClubsThe odds of that 1 card being the Ace of Clubs
(given that you already don’t hold it in your hand)(given that you already don’t hold it in your hand)
is 1/50, NOT 1/40!is 1/50, NOT 1/40!
 YOU ONLY KNOW 2 CARDS FOR SURE, SO THAT’SYOU ONLY KNOW 2 CARDS FOR SURE, SO THAT’S
ALL THE INFORMATION YOU CAN BASE YOURALL THE INFORMATION YOU CAN BASE YOUR
CALCULATION ON!CALCULATION ON!

16. A Simple ExampleA Simple Example
 Dealt:Dealt:
 The Flop:The Flop:
What is the ratio of outs if you areWhat is the ratio of outs if you are
going for 3 of a kind with 5’s?going for 3 of a kind with 5’s?

17. A Simple ExampleA Simple Example
 There are 2 remaining 5’s that canThere are 2 remaining 5’s that can
complete our 3 of a kind, so we have 2complete our 3 of a kind, so we have 2
outsouts
 There are 5 shown cards and 47There are 5 shown cards and 47
unseen cardsunseen cards
 Ratio of outs: 47:2 or 23.5:1Ratio of outs: 47:2 or 23.5:1

18. The Use of Pot Odds &The Use of Pot Odds &
OutsOuts
 Playing Texas Hold ‘EmPlaying Texas Hold ‘Em
 Dealt:Dealt:
 Raise $3 pre-FlopRaise $3 pre-Flop
 Both blinds fold, opponent on left callsBoth blinds fold, opponent on left calls
 Pot: $7.50, Flop:Pot: $7.50, Flop:

19. The Use of Pot Odds &The Use of Pot Odds &
OutsOuts
 You have the button, so you are the last to act afterYou have the button, so you are the last to act after
the flopthe flop
 Your opponent bets $7.50, doubling the pot to $15Your opponent bets $7.50, doubling the pot to $15
 You are going for a flush, do you call or fold?You are going for a flush, do you call or fold?
 Calculate the pot odds: $15 in the pot, have to putCalculate the pot odds: $15 in the pot, have to put
in $7.50 to call, so 15:7.5 or 2:1in $7.50 to call, so 15:7.5 or 2:1
 Calculate the ratio of outs: 4 diamonds that weCalculate the ratio of outs: 4 diamonds that we
know of, leaving 9 left that could help your hand toknow of, leaving 9 left that could help your hand to
get the flush. There are 47 unknown cards in total,get the flush. There are 47 unknown cards in total,
so 9 out of 47 cards can help, that’s 47:9 or 5.22:1so 9 out of 47 cards can help, that’s 47:9 or 5.22:1
 Since the ratio of outs is greater than the pot odds,Since the ratio of outs is greater than the pot odds,
you cannot profitably callyou cannot profitably call

20. Same problem done withSame problem done with
Mathematical ExpectationMathematical Expectation
 We have:We have:
– w = 15w = 15
– pw = 9/47pw = 9/47
– v = 7.5v = 7.5
– pl = 38/47pl = 38/47
E = (15*(E = (15*(99
//4747)) + (7.5*()) + (7.5*(3838
//4747)) = -3.191)) = -3.191
Negative mathematical expectation, so don’tNegative mathematical expectation, so don’t
call!call!

21. AlterationAlteration
 Say your opponent bets only $1, soSay your opponent bets only $1, so
you have to put in $1 to callyou have to put in $1 to call
 Calculate your pot odds: $8.50 in pot,Calculate your pot odds: $8.50 in pot,
$1 to call, so 8.5:1$1 to call, so 8.5:1
 Ratio of outs stays the same, so haveRatio of outs stays the same, so have
49:9 or 5.22:149:9 or 5.22:1
 Now your ratio of outs is less thanNow your ratio of outs is less than
your pot odds, thus you have ayour pot odds, thus you have a
positive expectation and should call!positive expectation and should call!

22.  Thousands of people with thousands ofThousands of people with thousands of
opinions about pokeropinions about poker
 Different ideas of how to become aDifferent ideas of how to become a
good poker player and what some ofgood poker player and what some of
the terms meanthe terms mean
– You might know different (and better)You might know different (and better)
information about Pokerinformation about Poker

23. The FundamentalThe Fundamental
Theorem of PokerTheorem of Poker
 ““Every time you play a hand differently fromEvery time you play a hand differently from
the way you would have played if you couldthe way you would have played if you could
see all your opponents’ cards, they gain;see all your opponents’ cards, they gain;
and every time you play your hand the sameand every time you play your hand the same
way you would have played it if you couldway you would have played it if you could
see their cards, they lose. Conversely,see their cards, they lose. Conversely,
every time opponents play their handsevery time opponents play their hands
differently from the way they could have ifdifferently from the way they could have if
they could see all your cards, you gain; andthey could see all your cards, you gain; and
every time they play their hands the sameevery time they play their hands the same
way they would have played if they couldway they would have played if they could
see your cards, you lose.see your cards, you lose.

24. The FundamentalThe Fundamental
Theorem of PokerTheorem of Poker
 What exactly does this mean?What exactly does this mean?
– Ex: Your opponent has pocket Aces andEx: Your opponent has pocket Aces and
you have a flush. If he were to see youryou have a flush. If he were to see your
hand, he would throw away his Aces, buthand, he would throw away his Aces, but
instead he calls.instead he calls.
 Calling was a mistake, but not a bad move, itCalling was a mistake, but not a bad move, it
was just played differently than if he knew whatwas just played differently than if he knew what
you hadyou had

25. More to PokerMore to Poker
““Knowing the mathematics of poker canKnowing the mathematics of poker can
certainly help you play a better game.certainly help you play a better game.
However, mathematics is only a smallHowever, mathematics is only a small
part of poker logic, and while it ispart of poker logic, and while it is
important, it is far less important thanimportant, it is far less important than
understanding and using theunderstanding and using the
underlying concepts of poker.”underlying concepts of poker.”

26. More to PokerMore to Poker
 PositionPosition
 BluffingBluffing
 Reading your opponents and knowingReading your opponents and knowing
their styletheir style
 Reading handsReading hands
 Slow playingSlow playing
 Loose and tight playLoose and tight play
 ..........