Mathematics and the Game of Poker

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

The History of Poker • Over the past 10 centuries poker has evolved from various games • 969 AD: Emperor Mutsung in China • 12th & 13th centuries: Eyptians • 16th century: “Primero” is often called “poker’s mother” • Each player was dealt 3 cards and bluffing was a very large part of the game

The History of Poker • In the U.S. • 1834: Being played on Mississippi Riverboats • Referred to as the “cheating game” • Civil War: extremely popular with soldiers for both the North and South • Wild West period: poker table found in a saloon in almost every town across the country

The Different Games of Poker • 5 Card Draw – grew in popularity after the Civil War and remained the most popular for almost a century • 7 Card Stud – shorty before WWII became the most popular and remained so for 40 years • Texas Hold ‘Em – became the dominant game in the 1970’s. Most prominent game of poker in the world. -hundreds of forms of poker exist

Basic Rules of Texas Hold ‘Em • The point of poker is to make money • “when the cards are dealt; you are no longer a grandson, a friend, or a nice guy; you are a player” (Sklyansky) • Post big blind and little blind • Dealer deals each player 2 cards face down • Betting begins – can call, raise, or fold • The Flop – the dealer burns the top card and places 3 cards on table face up. 2nd round of betting

Basic Rules of Texas Hold ‘Em • The Turn – burns a card and another card placed face up on table. 3rd round of betting • The River – burns a card and places the last card face up on table. 4th and final round of betting • A player can use any combination of the 7 available cards – 5 community cards and 2 in hand – to make best 5 card poker hand • Hands are revealed. The best hand wins.

Mathematical Expectation • Known as the expected value in Statistics, though name is misleading • Generally not a value that will be achieved • Better to think of it as the long term average value of the variable over numerous independent trials • In poker: the amount a bet will average winning or losing

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

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

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

Mathematical Expectation • w = $5 • pw = 1/5 • v = $1 • pl = 4/5 • E = (5 * 1/5)+(-1 * 4/5) = 1/5 =$0.20

Mathematical Expectation • In poker, it allows players to predict how much money they are going to win, or lose • The calculation of mathematical expectation, money management skills, and knowing the outs and pot odds allows a player to play a profitable game

Pot Odds & Outs • Outs: the number of cards left in the deck that will improve your hand • Ex: you have 4 spades on the Turn, so you have 9 outs left to get the flush on the River • Pot odds: the ratio of the amount of money in the pot to the bet you must call to continue in the hand • Ex: If there is currently $1000 in the pot and you have to put in $20 to call, your pot odds are 1000:20 or 50:1

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

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

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

Playing Texas Hold ‘Em Dealt: Raise $3 pre-Flop Both blinds fold, opponent on left calls Pot: $7.50, Flop: The Use of Pot Odds & Outs

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

Same problem done with Mathematical Expectation • We have: • w = 15 • pw = 9/47 • v = 7.5 • pl = 38/47 E = (15*(9/47)) + (7.5*(38/47)) = -3.191 Negative mathematical expectation, so don’t call!

Alteration • Say your opponent bets only $1, so you have to put in $1 to call • Calculate your pot odds: $8.50 in pot, $1 to call, so 8.5:1 • Ratio of outs stays the same, so have 49:9 or 5.22:1 • Now your ratio of outs is less than your pot odds, thus you have a positive expectation and should call!

Thousands of people with thousands of opinions about poker • Different ideas of how to become a good poker player and what some of the terms mean • You might know different (and better) information about Poker

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

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

The math for poker doesn’t stop there • http://www.learn-texas-holdem.com/texas-holdem-odds-probabilities.htm

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

More to Poker • Position • Bluffing • Reading your opponents and knowing their style • Reading hands • Slow playing • Loose and tight play • .....

Sources • The Theory of Poker by David Sklansky • www.poker.com • http://boardgames.about.com/cs/poker/a/texas_rules.htm • http://www.hundredpercentgambling.com/mathematical_expectation_of_a_bet.htm • http://wizardofodds.com/poker • http://www.pokerteam.com/mathematical-expectation.html • http://www.handsofpoker.net/poker-strategy/beginners-pot-odds • http://www.texasholdem-poker.com/odds_outs

Mathematics and the Game of Poker