Mathematics and the Game of Poker

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Mathematics and the Game of Poker
Kristina Fitzhugh 9/29/09
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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 pokers
  mother
• Each player was dealt 3 cards and bluffing was a
  very large part of the game

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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

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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
  1970s. Most prominent game of poker in the
  world.
• -hundreds of forms of poker exist

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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

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Basic Rules of Texas Hold Em

1. The Turn burns a card and another card placed
   face up on table. 3rd round of betting
2. The River burns a card and places the last card
   face up on table. 4th and final round of betting
3. 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
4. Hands are revealed. The best hand wins.

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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

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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

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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

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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?

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Mathematical Expectation

• w 5
• pw 1/5
• v 1
• pl 4/5
• E (5 1/5)(-1 4/5) 1/5 0.20

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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

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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
  100020 or 501

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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 dont hold it in your
  hand) is 1/50, NOT 1/40!
• YOU ONLY KNOW 2 CARDS FOR SURE, SO THATS ALL THE
  INFORMATION YOU CAN BASE YOUR CALCULATION ON!

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A Simple Example

• Dealt

• The Flop

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

• There are 2 remaining 5s 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 472 or 23.51

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The Use of Pot Odds Outs

• Playing Texas Hold Em
• Dealt

• Raise 3 pre-Flop
• Both blinds fold, opponent on left calls
• Pot 7.50, Flop

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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 157.5 or 21
• 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, thats 479
  or 5.221
• Since the ratio of outs is greater than the pot
  odds, you cannot profitably call

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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 dont call!

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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 8.51
• Ratio of outs stays the same, so have 499 or
  5.221
• Now your ratio of outs is less than your pot
  odds, thus you have a positive expectation and
  should call!

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• 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

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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.

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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

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• The math for poker doesnt stop there
• http//www.learn-texas-holdem.com/texas-holdem-odd
  s-probabilities.htm

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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.

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More to Poker

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

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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/beginne
  rs-pot-odds
• http//www.texasholdem-poker.com/odds_outs