Mathematics and the Game of Poker Kristina Fitzhugh 9/29/09

Mathematics and the Mathematics and the Game of PokerGame of Poker

Kristina Fitzhugh

9/29/09

The History of PokerThe History of Poker

Over the past 10 centuries Over the past 10 centuries poker has evolved from poker has evolved from various gamesvarious games– 969 AD: Emperor Mutsung in 969 AD: Emperor Mutsung in

China China – 1212thth & 13 & 13thth centuries: Eyptians centuries: Eyptians – 1616thth century: “Primero” is often century: “Primero” is often

called “poker’s mother” called “poker’s mother” Each player was dealt 3 cards and Each player was dealt 3 cards and

bluffing was a very large part of bluffing was a very large part of the gamethe game

The History of PokerThe History of Poker

In the U.S.In the U.S.– 1834: Being played on 1834: Being played on

Mississippi Riverboats Mississippi Riverboats Referred to as the Referred to as the

“cheating game”“cheating game”

― Civil War: extremely popular with soldiers Civil War: extremely popular with soldiers for both the North and Southfor both the North and South― Wild West period: poker table found in Wild West period: poker table found in a saloon in almost every town across the a saloon in almost every town across the countrycountry

The Different Games of The Different Games of PokerPoker

1)1) 5 Card Draw – grew in popularity after the 5 Card Draw – grew in popularity after the Civil War and remained the most popular Civil War and remained the most popular for almost a centuryfor almost a century

2)2) 7 Card Stud – shorty before WWII became 7 Card Stud – shorty before WWII became the most popular and remained so for 40 the most popular and remained so for 40 yearsyears

3)3) Texas Hold ‘Em – became the dominant Texas Hold ‘Em – became the dominant game in the 1970’s. Most prominent game game 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

Basic Rules of Basic 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 no when the cards are dealt; you are no

longer a grandson, a friend, or a nice longer 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 places – the dealer burns the top card and places 3 cards on table face up. 23 cards on table face up. 2ndnd round of betting round of betting

Basic Rules of Basic 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 betting round 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 final and final round of bettinground of betting

7)7) A player can use any combination of the 7 A player can use any combination of the 7 available cards – 5 community cards and 2 available 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.

Mathematical ExpectationMathematical Expectation

Known as the expected value in Known as the expected value in Statistics, though name is misleadingStatistics, though name is misleading

Generally not a value that will be Generally not a value that will be achievedachieved

Better to think of it as the long term Better to think of it as the long term average value of the variable over average value of the variable over numerous independent trialsnumerous independent trials

In poker: the amount a bet will In poker: the amount a bet will average winning or losingaverage winning or losing


Example: Example: – betting a friend $1 on the flip of a coin. Each time it betting a friend $1 on the flip of a coin. Each time it

comes up head, you win. Each time it comes up 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 flips Cannot 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


Now, say your friend (who is not too Now, say your friend (who is not too intelligent) wants to bet $2 to your $1 on the intelligent) 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


A person chooses a number between 1 A 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 you They 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?


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


In poker, it allows players to predict In poker, it allows players to predict how much money they are going to how much money they are going to win, or losewin, or lose

The calculation of mathematical The calculation of mathematical expectation, money management expectation, money management skills, and knowing the outs and pot skills, and knowing the outs and pot odds allows a player to play a odds allows a player to play a profitable gameprofitable game

Pot Odds & OutsPot Odds & Outs

Outs: the number of cards left in the Outs: 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 you Ex: you have 4 spades on the Turn, so you

have 9 outs left to get the flush on the have 9 outs left to get the flush on the RiverRiver

Pot odds: the ratio of the amount of Pot odds: the ratio of the amount of money in the pot to the bet you must money 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 pot Ex: If there is currently $1000 in the pot

and you have to put in $20 to call, your and you have to put in $20 to call, your pot odds are 1000:20 or 50:1pot odds are 1000:20 or 50:1

Odds with Exposed & Odds with Exposed & Unseen CardsUnseen Cards

When figuring the outs, why are the When figuring the outs, why are the burned cards and the number of cards burned 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 remaining You 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 Clubs The 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’S YOU ONLY KNOW 2 CARDS FOR SURE, SO THAT’S ALL THE INFORMATION YOU CAN BASE YOUR ALL THE INFORMATION YOU CAN BASE YOUR CALCULATION ON!CALCULATION ON!

A Simple ExampleA Simple Example

Dealt: Dealt:

The Flop: The Flop:

What is the ratio of outs if you are going What is the ratio of outs if you are going for 3 of a kind with 5’s?for 3 of a kind with 5’s?

A Simple ExampleA Simple Example

There are 2 remaining 5’s that can There are 2 remaining 5’s that can complete our 3 of a kind, so we have 2 complete our 3 of a kind, so we have 2 outsouts

There are 5 shown cards and 47 There 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

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:

The Use of Pot Odds & The Use of Pot Odds & OutsOuts

You have the button, so you are the last to act You have the button, so you are the last to act after the flopafter the flop

Your opponent bets $7.50, doubling the pot to Your opponent bets $7.50, doubling the pot to $15$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 Calculate the pot odds: $15 in the pot, have to

put in $7.50 to call, so 15:7.5 or 2:1put in $7.50 to call, so 15:7.5 or 2:1 Calculate the ratio of outs: 4 diamonds that we Calculate the ratio of outs: 4 diamonds that we

know of, leaving 9 left that could help your hand know of, leaving 9 left that could help your hand to get the flush. There are 47 unknown cards in to get the flush. There are 47 unknown cards in total, so 9 out of 47 cards can help, that’s 47:9 or total, so 9 out of 47 cards can help, that’s 47:9 or 5.22:15.22:1

Since the ratio of outs is greater than the pot Since the ratio of outs is greater than the pot odds, you cannot profitably callodds, you cannot profitably call

Same problem done with Same 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 Negative mathematical expectation, so don’t call!don’t call!

AlterationAlteration

Say your opponent bets only $1, so Say 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 have Ratio 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 than your Now your ratio of outs is less than your pot odds, thus you have a positive pot odds, thus you have a positive expectation and should call!expectation and should call!

Thousands of people with thousands of Thousands of people with thousands of opinions about pokeropinions about poker

Different ideas of how to become a Different ideas of how to become a good poker player and what some of good 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

The Fundamental The Fundamental Theorem of PokerTheorem of Poker

““Every time you play a hand differently Every time you play a hand differently from the way you would have played if from the way you would have played if you could see all your opponents’ cards, you could see all your opponents’ cards, they gain; and every time you play your they gain; and every time you play your hand the same way you would have hand the same way you would have played it if you could see their cards, played it if you could see their cards, they lose. Conversely, every time they lose. Conversely, every time opponents play their hands differently opponents play their hands differently from the way they could have if they from the way they could have if they could see all your cards, you gain; and could see all your cards, you gain; and every time they play their hands the every time they play their hands the same way they would have played if same way they would have played if they could see your cards, you lose.they could see your cards, you lose.

The Fundamental The Fundamental Theorem of PokerTheorem of Poker

What exactly does this mean?What exactly does this mean?– Ex: Your opponent has pocket Aces and Ex: Your opponent has pocket Aces and

you have a flush. If he were to see your you have a flush. If he were to see your hand, he would throw away his Aces, but hand, he would throw away his Aces, but instead he calls. instead he calls.

Calling was a mistake, but not a bad move, it Calling was a mistake, but not a bad move, it was just played differently than if he knew what was just played differently than if he knew what you hadyou had

The math for poker doesn’t stop thereThe math for poker doesn’t stop there http://www.learn-texas-holdem.com/http://www.learn-texas-holdem.com/

texas-holdem-odds-probabilities.htmtexas-holdem-odds-probabilities.htm

More to PokerMore to Poker

““Knowing the mathematics of poker can Knowing the mathematics of poker can certainly help you play a better game. certainly help you play a better game. However, mathematics is only a small However, mathematics is only a small part of poker logic, and while it is part of poker logic, and while it is important, it is far less important than important, it is far less important than understanding and using the understanding and using the underlying concepts of poker.”underlying concepts of poker.”

More to PokerMore to Poker

PositionPosition BluffingBluffing Reading your opponents and knowing Reading your opponents and knowing

their styletheir style Reading handsReading hands Slow playingSlow playing Loose and tight playLoose and tight play ..........

SourcesSources

The Theory of PokerThe Theory of Poker by David Sklansky by David Sklansky www.poker.comwww.poker.com http://boardgames.about.com/cs/poker/a/texas_rules.htmhttp://boardgames.about.com/cs/poker/a/texas_rules.htm http://www.hundredpercentgambling.com/http://www.hundredpercentgambling.com/

mathematical_expectation_of_a_bet.htmmathematical_expectation_of_a_bet.htm http://wizardofodds.com/pokerhttp://wizardofodds.com/poker http://www.pokerteam.com/mathematical-expectation.htmlhttp://www.pokerteam.com/mathematical-expectation.html http://www.handsofpoker.net/poker-strategy/beginners-pot-http://www.handsofpoker.net/poker-strategy/beginners-pot-

oddsodds http://www.texasholdem-poker.com/odds_outshttp://www.texasholdem-poker.com/odds_outs

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Mathematics and the Game of Poker Kristina Fitzhugh 9/29/09