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Introduction to Discrete Probability. Epp, section 6. x CS 202 Aaron Bloomfield. Terminology. Experiment A repeatable procedure that yields one of a given set of outcomes Rolling a die, for example Sample space The range of outcomes possible For a die, that would be values 1 to 6 Event

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introduction to discrete probability

Introduction to Discrete Probability

Epp, section 6.x

CS 202

Aaron Bloomfield

terminology
Terminology
  • Experiment
    • A repeatable procedure that yields one of a given set of outcomes
    • Rolling a die, for example
  • Sample space
    • The range of outcomes possible
    • For a die, that would be values 1 to 6
  • Event
    • One of the sample outcomes that occurred
    • If you rolled a 4 on the die, the event is the 4
probability definition
Probability definition
  • The probability of an event occurring is:
    • Where E is the set of desired events (outcomes)
    • Where S is the set of all possible events (outcomes)
    • Note that 0 ≤ |E| ≤ |S|
      • Thus, the probability will always between 0 and 1
      • An event that will never happen has probability 0
      • An event that will always happen has probability 1
probability is always a value between 0 and 1
Probability is always a value between 0 and 1
  • Something with a probability of 0 will never occur
  • Something with a probability of 1 will always occur
  • You cannot have a probability outside this range!
  • Note that when somebody says it has a “100% probability)
    • That means it has a probability of 1
dice probability
Dice probability
  • What is the probability of getting “snake-eyes” (two 1’s) on two six-sided dice?
    • Probability of getting a 1 on a 6-sided die is 1/6
    • Via product rule, probability of getting two 1’s is the probability of getting a 1 AND the probability of getting a second 1
    • Thus, it’s 1/6 * 1/6 = 1/36
  • What is the probability of getting a 7 by rolling two dice?
    • There are six combinations that can yield 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
    • Thus, |E| = 6, |S| = 36, P(E) = 6/36 = 1/6
the game of poker
The game of poker
  • You are given 5 cards (this is 5-card stud poker)
  • The goal is to obtain the best hand you can
  • The possible poker hands are (in increasing order):
    • No pair
    • One pair (two cards of the same face)
    • Two pair (two sets of two cards of the same face)
    • Three of a kind (three cards of the same face)
    • Straight (all five cards sequentially – ace is either high or low)
    • Flush (all five cards of the same suit)
    • Full house (a three of a kind of one face and a pair of another face)
    • Four of a kind (four cards of the same face)
    • Straight flush (both a straight and a flush)
    • Royal flush (a straight flush that is 10, J, K, Q, A)
poker probability royal flush
Poker probability: royal flush
  • What is the chance ofgetting a royal flush?
    • That’s the cards 10, J, Q, K, and A of the same suit
  • There are only 4 possible royal flushes
  • Possibilities for 5 cards: C(52,5) = 2,598,960
  • Probability = 4/2,598,960 = 0.0000015
    • Or about 1 in 650,000
poker probability four of a kind
Poker probability: four of a kind
  • What is the chance of getting 4 of a kind when dealt 5 cards?
    • Possibilities for 5 cards: C(52,5) = 2,598,960
  • Possible hands that have four of a kind:
    • There are 13 possible four of a kind hands
    • The fifth card can be any of the remaining 48 cards
    • Thus, total possibilities is 13*48 = 624
  • Probability = 624/2,598,960 = 0.00024
    • Or 1 in 4165
poker probability flush
Poker probability: flush
  • What is the chance of getting a flush?
    • That’s all 5 cards of the same suit
  • We must do ALL of the following:
    • Pick the suit for the flush: C(4,1)
    • Pick the 5 cards in that suit: C(13,5)
  • As we must do all of these, we multiply the values out (via the product rule)
  • This yields
  • Possibilities for 5 cards: C(52,5) = 2,598,960
  • Probability = 5148/2,598,960 = 0.00198
    • Or about 1 in 505
  • Note that if you don’t count straight flushes (and thus royal flushes) as a “flush”, then the number is really 5108
poker probability full house
Poker probability: full house
  • What is the chance of getting a full house?
    • That’s three cards of one face and two of another face
  • We must do ALL of the following:
    • Pick the face for the three of a kind: C(13,1)
    • Pick the 3 of the 4 cards to be used: C(4,3)
    • Pick the face for the pair: C(12,1)
    • Pick the 2 of the 4 cards of the pair: C(4,2)
  • As we must do all of these, we multiply the values out (via the product rule)
  • This yields
  • Possibilities for 5 cards: C(52,5) = 2,598,960
  • Probability = 3744/2,598,960 = 0.00144
    • Or about 1 in 694
inclusion exclusion principle
Inclusion-exclusion principle
  • The possible poker hands are (in increasing order):
    • Nothing
    • One pair cannot include two pair, three of a kind, four of a kind, or full house
    • Two pair cannot include three of a kind, four of a kind, or full house
    • Three of a kind cannot include four of a kind or full house
    • Straight cannot include straight flush or royal flush
    • Flush cannot include straight flush or royal flush
    • Full house
    • Four of a kind
    • Straight flush cannot include royal flush
    • Royal flush
poker probability three of a kind
Poker probability: three of a kind
  • What is the chance of getting a three of a kind?
    • That’s three cards of one face
    • Can’t include a full house or four of a kind
  • We must do ALL of the following:
    • Pick the face for the three of a kind: C(13,1)
    • Pick the 3 of the 4 cards to be used: C(4,3)
    • Pick the two other cards’ face values: C(12,2)
      • We can’t pick two cards of the same face!
    • Pick the suits for the two other cards: C(4,1)*C(4,1)
  • As we must do all of these, we multiply the values out (via the product rule)
  • This yields
  • Possibilities for 5 cards: C(52,5) = 2,598,960
  • Probability = 54,912/2,598,960 = 0.0211
    • Or about 1 in 47
poker hand odds
Poker hand odds
  • The possible poker hands are (in increasing order):
    • Nothing 1,302,540 0.5012
    • One pair 1,098,240 0.4226
    • Two pair 123,552 0.0475
    • Three of a kind 54,912 0.0211
    • Straight 10,200 0.00392
    • Flush 5,108 0.00197
    • Full house 3,744 0.00144
    • Four of a kind 624 0.000240
    • Straight flush 36 0.0000139
    • Royal flush 4 0.00000154
more on probabilities
More on probabilities
  • Let E be an event in a sample space S. The probability of the complement of E is:
  • Recall the probability for getting a royal flush is 0.0000015
    • The probability of not getting a royal flush is 1-0.0000015 or 0.9999985
  • Recall the probability for getting a four of a kind is 0.00024
    • The probability of not getting a four of a kind is 1-0.00024 or 0.99976
probability of the union of two events
Probability of the union of two events
  • Let E1 and E2 be events in sample space S
  • Then p(E1 U E2) = p(E1) + p(E2) – p(E1 ∩ E2)
  • Consider a Venn diagram dart-board
probability of the union of two events2
Probability of the union of two events
  • If you choose a number between 1 and 100, what is the probability that it is divisible by 2 or 5 or both?
  • Let n be the number chosen
    • p(2|n) = 50/100 (all the even numbers)
    • p(5|n) = 20/100
    • p(2|n) and p(5|n) = p(10|n) = 10/100
    • p(2|n) or p(5|n) = p(2|n) + p(5|n) - p(10|n)

= 50/100 + 20/100 – 10/100

= 3/5

when is gambling worth it
When is gambling worth it?
  • This is a statistical analysis, not a moral/ethical discussion
  • What if you gamble $1, and have a ½ probability to win $10?
    • If you play 100 times, you will win (on average) 50 of those times
      • Each play costs $1, each win yields $10
      • For $100 spent, you win (on average) $500
    • Average win is $5 (or $10 * ½) per play for every $1 spent
  • What if you gamble $1 and have a 1/100 probability to win $10?
    • If you play 100 times, you will win (on average) 1 of those times
      • Each play costs $1, each win yields $10
      • For $100 spent, you win (on average) $10
    • Average win is $0.10 (or $10 * 1/100) for every $1 spent
  • One way to determine if gambling is worth it:
    • probability of winning * payout ≥ amount spent
    • Or p(winning) * payout ≥ investment
    • Of course, this is a statistical measure
when is lotto worth it
When is lotto worth it?
  • Many older lotto games you have to choose 6 numbers from 1 to 48
    • Total possible choices is C(48,6) = 12,271,512
    • Total possible winning numbers is C(6,6) = 1
    • Probability of winning is 0.0000000814
      • Or 1 in 12.3 million
  • If you invest $1 per ticket, it is only statistically worth it if the payout is > $12.3 million
    • As, on the “average” you will only make money that way
    • Of course, “average” will require trillions of lotto plays…
powerball lottery
Powerball lottery
  • Modern powerball lottery is a bit different
    • Source: http://en.wikipedia.org/wiki/Powerball
  • You pick 5 numbers from 1-55
    • Total possibilities: C(55,5) = 3,478,761
  • You then pick one number from 1-42 (the powerball)
    • Total possibilities: C(42,1) = 42
  • By the product rule, you need to do both
    • So the total possibilities is 3,478,761* 42 = 146,107,962
  • While there are many “sub” prizes, the probability for the jackpot is about 1 in 146 million
    • You will “break even” if the jackpot is $146M
    • Thus, one should only play if the jackpot is greater than $146M
  • If you count in the other prizes, then you will “break even” if the jackpot is $121M
blackjack1
Blackjack
  • You are initially dealt two cards
    • 10, J, Q and K all count as 10
    • Ace is EITHER 1 or 11 (player’s choice)
  • You can opt to receive more cards (a “hit”)
  • You want to get as close to 21 as you can
    • If you go over, you lose (a “bust”)
  • You play against the house
    • If the house has a higher score than you, then you lose
blackjack probabilities
Blackjack probabilities
  • Getting 21 on the first two cards is called a blackjack
    • Or a “natural 21”
  • Assume there is only1 deck of cards
  • Possible blackjack blackjack hands:
    • First card is an A, second card is a 10, J, Q, or K
      • 4/52 for Ace, 16/51 for the ten card
      • = (4*16)/(52*51) = 0.0241 (or about 1 in 41)
    • First card is a 10, J, Q, or K; second card is an A
      • 16/52 for the ten card, 4/51 for Ace
      • = (16*4)/(52*51) = 0.0241 (or about 1 in 41)
  • Total chance of getting a blackjack is the sum of the two:
    • p = 0.0483, or about 1 in 21
    • How appropriate!
    • More specifically, it’s 1 in 20.72 (0.048)
blackjack probabilities1
Blackjack probabilities
  • Another way to get 20.72
  • There are C(52,2) = 1,326 possible initial blackjack hands
  • Possible blackjack blackjack hands:
    • Pick your Ace: C(4,1)
    • Pick your 10 card: C(16,1)
    • Total possibilities is the product of the two (64)
  • Probability is 64/1,326 = 1 in 20.72 (0.048)
blackjack probabilities2
Blackjack probabilities
  • Getting 21 on the first two cards is called a blackjack
  • Assume there is an infinite deck of cards
    • So many that the probably of getting a given card is not affected by any cards on the table
  • Possible blackjack blackjack hands:
    • First card is an A, second card is a 10, J, Q, or K
      • 4/52 for Ace, 16/52 for second part
      • = (4*16)/(52*52) = 0.0236 (or about 1 in 42)
    • First card is a 10, J, Q, or K; second card is an A
      • 16/52 for first part, 4/52 for Ace
      • = (16*4)/(52*52) = 0.0236 (or about 1 in 42)
  • Total chance of getting a blackjack is the sum:
    • p = 0.0473, or about 1 in 21
    • More specifically, it’s 1 in 21.13 (vs. 20.72)
  • In reality, most casinos use “shoes” of 6-8 decks for this reason
    • It slightly lowers the player’s chances of getting a blackjack
    • And prevents people from counting the cards…
counting cards and continuous shuffling machines csms
Counting cards and Continuous Shuffling Machines (CSMs)
  • Counting cards means keeping track of which cards have been dealt, and how that modifies the chances
    • There are “easy” ways to do this – count all aces and 10-cards instead of all cards
  • Yet another way for casinos to get the upper hand
    • It prevents people from counting the “shoes” of 6-8 decks of cards
  • After cards are discarded, they are added to the continuous shuffling machine
  • Many blackjack players refuse to play at a casino with one
    • So they aren’t used as much as casinos would like
so always use a single deck right
So always use a single deck, right?
  • Most people think that a single-deck blackjack table is better, as the player’s odds increase
    • And you can try to count the cards
  • But it’s usually not the case!
  • Normal rules have a 3:2 payout for a blackjack
    • If you bet $100, you get your $100 back plus 3/2 * $100, or $150 additional
  • Most single-deck tables have a 6:5 payout
    • You get your $100 back plus 6/5 * $100 or $120 additional
    • This lowered benefit of being able to count the cards OUTWEIGHSthe benefit of the single deck!
      • And thus the benefit of counting the cards
      • Even with counting cards
    • You cannot win money on a 6:5 blackjack table that uses 1 deck
    • Remember, the house always wins
blackjack probabilities when to hold
Blackjack probabilities: when to hold
  • House usually holds on a 17
    • What is the chance of a bust if you draw on a 17? 16? 15?
  • Assume all cards have equal probability
  • Bust on a draw on a 18
    • 4 or above will bust: that’s 10 (of 13) cards that will bust
    • 10/13 = 0.769 probability to bust
  • Bust on a draw on a 17
    • 5 or above will bust: 9/13 = 0.692 probability to bust
  • Bust on a draw on a 16
    • 6 or above will bust: 8/13 = 0.615 probability to bust
  • Bust on a draw on a 15
    • 7 or above will bust: 7/13 = 0.538 probability to bust
  • Bust on a draw on a 14
    • 8 or above will bust: 6/13 = 0.462 probability to bust
buying blackjack insurance
Buying (blackjack) insurance
  • If the dealer’s visible card is an Ace, the player can buy insurance against the dealer having a blackjack
    • There are then two bets going: the original bet and the insurance bet
    • If the dealer has blackjack, you lose your original bet, but your insurance bet pays 2-to-1
      • So you get twice what you paid in insurance back
      • Note that if the player also has a blackjack, it’s a “push”
    • If the dealer does not have blackjack, you lose your insurance bet, but your original bet proceeds normal
  • Is this insurance worth it?
buying blackjack insurance1
Buying (blackjack) insurance
  • If the dealer shows an Ace, there is a 4/13 = 0.308 probability that they have a blackjack
    • Assuming an infinite deck of cards
    • Any one of the “10” cards will cause a blackjack
  • If you bought insurance 1,000 times, it would be used 308 (on average) of those times
    • Let’s say you paid $1 each time for the insurance
  • The payout on each is 2-to-1, thus you get $2 back when you use your insurance
    • Thus, you get 2*308 = $616 back for your $1,000 spent
  • Or, using the formula p(winning) * payout ≥ investment
    • 0.308 * $2 ≥ $1
    • 0.616 ≥ $1
    • Thus, it’s not worth it
  • Buying insurance is considered a very poor option for the player
    • Hence, almost every casino offers it
blackjack strategy
Blackjack strategy
  • These tables tell you the best move to do on each hand
  • The odds are still (slightly) in the house’s favor
  • The house always wins…
why counting cards doesn t work well
Why counting cards doesn’t work well…
  • If you make two or three mistakes an hour, you lose any advantage
    • And, in fact, cause a disadvantage!
  • You lose lots of money learning to count cards
  • Then, once you can do so, you are banned from the casinos
so why is blackjack so popular
So why is Blackjack so popular?
  • Although the casino has the upper hand, the odds are much closer to 50-50 than with other games
    • Notable exceptions are games that you are not playing against the house – i.e., poker
      • You pay a fixed amount per hand
as seen in a casino
As seen ina casino
  • This wheel is spun if:
    • You place $1 on the “spin the wheel” square
    • You get a natural blackjack
    • You lose the dollar either way
  • You win the amount shown on the wheel
is it worth it to place 1 on the square
Is it worth it to place $1 on the square?
  • The amounts on the wheel are:
    • 30, 1000, 11, 20, 16, 40, 15, 10, 50, 12, 25, 14
    • Average is $103.58
  • Chance of a natural blackjack:
    • p = 0.0473, or 1 in 21.13
  • So use the formula:
    • p(winning) * payout ≥ investment
    • 0.0473 * $103.58 ≥ $1
    • $4.90 ≥ $1
    • But the house always wins! So what happened?
as seen in a casino1
As seen ina casino
  • Note that not all amounts have an equal chance of winning
    • There are 2 spots to win $15
    • There is ONE spot to win $1,000
    • Etc.
back to the drawing board
Back to the drawing board
  • If you weight each “spot” by the amount it can win, you get $1609 for 30 “spots”
    • That’s an average of $53.63 per spot
  • So use the formula:
    • p(winning) * payout ≥ investment
    • 0.0473 * $53.63 ≥ $1
    • $2.54 ≥ $1
    • Still not there yet…
my theory
My theory
  • I think the wheel is weighted so the $1,000 side of the wheel is heavy and thus won’t be chosen
    • As the “chooser” is at the top
    • But I never saw it spin, so I can’t say for sure
  • Take the $1,000 out of the 30 spot discussion of the last slide
    • That leaves $609 for 29 spots
    • Or $21.00 per spot
  • So use the formula:
    • p(winning) * payout ≥ investment
    • 0.0473 * $21 ≥ $1
    • $0.9933 ≥ $1
  • And I’m probably still missing something here…
  • Remember that the house always wins!
roulette1
Roulette
  • A wheel with 38 spots is spun
    • Spots are numbered 1-36, 0, and 00
    • European casinos don’t have the 00
  • A ball drops into one of the 38 spots
  • A bet is placed as to which spot or spots the ball will fall into
    • Money is then paid out if the ball lands in the spot(s) you bet upon
the roulette table1
The Roulette table
  • Bets can be placed on:
    • A single number
    • Two numbers
    • Four numbers
    • All even numbers
    • All odd numbers
    • The first 18 nums
    • Red numbers

Probability:

1/38

2/38

4/38

18/38

18/38

18/38

18/38

the roulette table2
The Roulette table
  • Bets can be placed on:
    • A single number
    • Two numbers
    • Four numbers
    • All even numbers
    • All odd numbers
    • The first 18 nums
    • Red numbers

Probability:

1/38

2/38

4/38

18/38

18/38

18/38

18/38

Payout:

36x

18x

9x

2x

2x

2x

2x

roulette2
Roulette
  • It has been proven that proven that no advantageous strategies exist
  • Including:
    • Learning the wheel’s biases
      • Casino’s regularly balance their Roulette wheels
    • Using lasers (yes, lasers) to check the wheel’s spin
      • What casino will let you set up a laser inside to beat the house?
roulette3
Roulette
  • It has been proven that proven that no advantageous strategies exist
  • Including:
    • Martingale betting strategy
      • Where you double your bet each time (thus making up for all previous losses)
      • It still won’t work!
      • You can’t double your money forever
        • It could easily take 50 times to achieve finally win
        • If you start with $1, then you must put in $1*250 = $1,125,899,906,842,624 to win this way!
        • That’s 1quadrillion
      • See http://en.wikipedia.org/wiki/Martingale_(roulette_system) for more info
what s behind door number three
What’s behind door number three?
  • The Monty Hall problem paradox
    • Consider a game show where a prize (a car) is behind one of three doors
    • The other two doors do not have prizes (goats instead)
    • After picking one of the doors, the host (Monty Hall) opens a different door to show you that the door he opened is not the prize
    • Do you change your decision?
  • Your initial probability to win (i.e. pick the right door) is 1/3
  • What is your chance of winning if you change your choice after Monty opens a wrong door?
  • After Monty opens a wrong door, if you change your choice, your chance of winning is 2/3
    • Thus, your chance of winning doubles if you change
    • Huh?
dealing cards
Dealing cards
  • Consider a dealt hand of cards
    • Assume they have not been seen yet
    • What is the chance of drawing a flush?
    • Does that chance change if I speak words after the experiment has completed?
    • Does that chance change if I tell you more info about what’s in the deck?
  • No!
    • Words spoken after an experiment has completed do not change the chance of an event happening by that experiment
      • No matter what is said
what s behind door number one hundred
What’s behind door number one hundred?
  • Consider 100 doors
    • You choose one
    • Monty opens 98 wrong doors
    • Do you switch?
  • Your initial chance of being right is 1/100
  • Right before your switch, your chance of being right is still 1/100
    • Just because you know more info about the other doors doesn’t change your chances
      • You didn’t know this info beforehand!
  • Your final chance of being right is 99/100 if you switch
    • You have two choices: your original door and the new door
    • The original door still has 1/100 chance of being right
    • Thus, the new door has 99/100 chance of being right
    • The 98 doors that were opened were not chosen at random!
      • Monty Hall knows which door the car is behind
  • Reference: http://en.wikipedia.org/wiki/Monty_Hall_problem
an aside probability of multiple events
An aside: probability of multiple events
  • Assume you have a 5/6 chance for an event to happen
    • Rolling a 1-5 on a die, for example
  • What’s the chance of that event happening twice in a row?
  • Cases:
    • Event happening neither time: 1/6 * 1/6 = 1/36
    • Event happening first time: 5/6 * 1/6 = 5/36
    • Event happening second time: 1/6 * 5/6 = 5/36
    • Event happening both times: 5/6 * 5/6 = 25/36
  • For an event to happen twice, the probabilityis the product of the individual probabilities
an aside probability of multiple events1
An aside: probability of multiple events
  • Assume you have a 5/6 chance for an event to happen
    • Rolling a 1-5 on a die, for example
  • What’s the chance of that event happening at least once?
  • Cases:
    • Event happening neither time: 1/6 * 1/6 = 1/36
    • Event happening first time: 5/6 * 1/6 = 5/36
    • Event happening second time: 1/6 * 5/6 = 5/36
    • Event happening both times: 5/6 * 5/6 = 25/36
  • It’s 35/36!
  • For an event to happen at least once, it’s 1 minus the probability of it never happening
  • Or 1 minus the compliment of it never happening
probability vs odds
Probability vs. odds
  • Consider an event that has a 1 in 3 chance of happening
  • Probability is 0.333
  • Which is a 1 in 3 chance
  • Or 2:1 odds
    • Meaning if you play it 3 (2+1) times, you will lose 2 times for every 1 time you win
  • This, if you have x:y odds, you probability is y/(x+y)
    • The y is usually 1, and the x is scaled appropriately
    • For example 2.2:1
      • That probability is 1/(1+2.2) = 1/3.2 = 0.313
  • 1:1 odds means that you will lose as many times as you win
texas hold em

Texas Hold’em

Reference:

http://teamfu.freeshell.org/poker_odds.html

texas hold em1
Texas Hold’em
  • The most popular poker variant today
  • Every player starts with two face down cards
    • Called “hole” or “pocket” cards
    • Hence the term “ace in the hole”
  • Five cards are placed in the center of the table
    • These are common cards, shared by every player
    • Initially they are placed face down
    • The first 3 cards are then turned face up, then the fourth card, then the fifth card
    • You can bet between the card turns
  • You try to make the best 5-card hand of the seven cards available to you
    • Your two hole cards and the 5 common cards
texas hold em2
Texas Hold’em
  • Hand progression
    • Note that anybody can fold at any time
    • Cards are dealt: 2 “hole” cards per player
    • 5 community cards are dealt face down (how this is done varies)
    • Bets are placed based on your pocket cards
    • The first three community cards are turned over (or dealt)
      • Called the “flop”
    • Bets are placed
    • The next community card is turned over (or dealt)
      • Called the “turn”
    • Bets are placed
    • The last community card is turned over (or dealt)
      • Called the “river”
    • Bets are placed
    • Hands are then shown to determine who wins the pot
texas hold em terminology
Texas Hold’em terminology
  • Pocket: your two face-down cards
  • Pocket pair: when you have a pair in your pocket
  • Flop: when the initial 3 community cards are shown
  • Turn: when the 4thcommunity card is shown
  • River: when the 5th community card is shown
  • Nuts (or nut hand): the best possible hand that you can hope for with the cards you have and the not-yet-shown cards
  • Outs: the number of cards you need to achieve your nut hand
  • Pot: the money in the center that is being bet upon
  • Fold: when you stop betting on the current hand
  • Call: when you match the current bet
odds of a texas hold em hand
Odds of a Texas Hold’em hand
  • Pick any poker hand
    • We’ll choose a royal flush
    • There are only 4 possibilities (1 of each suit)
  • There are 7 cards dealt
    • Total of C(52,7) = 133,784,560 possibilities
  • Chance of getting that in a Texas Hold’em game:
    • Choose the 5 cards of your royal flush: C(4,1)
    • Choose the remaining two cards: C(47,2)
    • Product rule: multiply them together
  • Result is 4324 (of 133,784,560) possibilities
    • Or 1 in 30,940
    • Or probability of 0.000,032
    • This is much more common than 1 in 649,740 for stud poker!
  • But nobody does Texas Hold’em probability that way, though…
an example of a hand using texas hold em terminology
An example of a hand usingTexas Hold’em terminology
  • Your pocket hand is J♥, 4♥
  • The flop shows 2♥, 7♥, K♣
  • There are two cards still to be revealed (the turn and the river)
  • Your nut hand is going to be a flush
    • As that’s the best hand you can (realistically) hope for with the cards you have
  • There are 9 cards that will allow you to achieve your flush
    • Any other heart
    • Thus, you have 9 outs
continuing with that example
Continuing with that example
  • There are 47 unknown cards
    • The two unturned cards, the other player’s cards, and the rest of the deck
  • There are 9 outs (the other 9 hearts)
  • What’s the chance you will get your flush?
    • Rephrased: what’s the chance that you will get an out on at least one of the remaining cards?
    • For an event to happen at least once, it’s 1 minus the probability of it never happening
    • Chances:
      • Out on neither turn nor river 38/47 * 37/46 = 0.65
      • Out on turn only 9/47 * 38/46 = 0.16
      • Out on river only 38/47 * 9/46 = 0.16
      • Out on both turn and river 9/47 * 8/46 = 0.03
    • All the chances add up to 1, as expected
    • Chance of getting at least 1 out is 1 minus the chance of not getting any outs
      • Or 1-0.65 = 0.35
      • Or 1 in 2.9
      • Or 1.9:1
continuing with that example1
Continuing with that example
  • What if you miss your out on the turn
  • Then what is the chance you will hit the out on the river?
  • There are 46 unknown cards
    • The two unturned cards, the other player’s cards, and the rest of the deck
  • There are still 9 outs (the other 9 hearts)
  • What’s the chance you will get your flush?
    • 9/46 = 0.20
    • Or 1 in 5.1
    • Or 4.1:1
    • The odds have significantly decreased!
  • These odds are called the hand odds
    • I.e. the chance that you will get your nut hand
hand odds vs pot odds
Hand odds vs. pot odds
  • So far we’ve seen the odds of getting a given hand
  • Assume that you are playing with only one other person
  • If you win the pot, you get a payout of two times what you invested
    • As you each put in half the pot
    • This is called the pot odds
      • Well, almost – we’ll see more about pot odds in a bit
  • After the flop, assume that the pot has $20, the bet is $10, and thus the call is $10
    • Payout (if you match the bet and then win) is $40
    • Your investment is $10
    • Your pot odds are 30:10 (not 40:10, as your call is not considered as part of the odds)
      • Or 3:1
  • When is it worth it to continue?
    • What if you have 3:1 hand odds (0.25 probability)?
    • What if you have 2:1 hand odds (0.33 probability)?
    • What if you have 1:1 hand odds (0.50 probability)?
  • Note that we did not consider the probabilities before the flop
hand odds vs pot odds1
Hand odds vs. pot odds
  • Pot payout is $40, investment is $10
  • Use the formula: p(winning) * payout ≥ investment
  • When is it worth it to continue?
    • We are assuming that your nut hand will win
      • A safe assumption for a flush, but not a tautology!
    • What if you have 3:1 hand odds (0.25 probability)?
      • 0.25 * $40 ≥ $10
      • $10 =$10
      • If you pursue this hand, you will make as much as you lose
    • What if you have 2:1 hand odds (0.33 probability)?
      • 0.33 * $40 ≥ $10
      • $13.33 > $10
      • Definitely worth it to continue!
    • What if you have 1:1 hand odds (0.50 probability)?
      • 0.5 * $40 ≥ $10
      • $20 >$10
      • Definitely worth it to continue!
pot odds
Pot odds
  • Pot odds is the ratio of the amount in the pot to the amount you have to call
  • In other words, we don’t consider any previously invested money
    • Only the current amount in the pot and the current amount of the call
    • The reason is that you are considering each bet as it is placed, not considering all of your (past and present) bets together
    • If you considered all the amounts invested, you must then consider the probabilities at each point that you invested money
    • Instead, we just take a look at each investment individually
    • Technically, these are mathematically equal, but the latter is much easier (and thus more realistic to do in a game)
  • In the last example, the pot odds were 3:1
    • As there was $30 in the pot, and the call was $10
      • Even though you invested some money previously
another take on pot odds
Another take on pot odds
  • Assume the pot is $100, and the call is $10
    • Thus, the pot odds are 100:10 or 10:1
    • You invest $10, and get $110 if you win
    • Thus, you have to win 1 out of 11 times to break even
    • Or have odds of 10:1
    • If you have better odds, you’ll make money in the long run
    • If you have worse odds, you’ll lose money in the long run
hand odds vs pot odds2
Hand odds vs. pot odds
  • Pot is now $20, investment is $10
    • Pot odds are thus 2:1
  • Use the formula: p(winning) * payout ≥ investment
  • When is it worth it to continue?
    • What if you have 3:1 hand odds (0.25 probability)?
      • 0.25 * $30 ≥ $10
      • $7.50 <$10
    • What if you have 2:1 hand odds (0.33 probability)?
      • 0.33 * $30 ≥ $10
      • $10 = $10
      • If you pursue this hand, you will make as much as you lose
    • What if you have 1:1 hand odds (0.50 probability)?
      • 0.5 * $30 ≥ $10
      • $15 >$10
  • The only time it is worth it to continue is when the pot odds outweigh the hand odds
    • Meaning the first part of the pot odds is greater than the first part of the hand odds
    • If you do not follow this rule, you will lose money in the long run
computing hand odds vs pot odds
Computing hand odds vs. pot odds
  • Consider the following hand progression:
  • Your hand: almost a flush (4 out of 5 cards of one suit)
    • Called a “flush draw”
      • Perhaps because one more draw can make it a flush
  • On the flop: $5 pot, $10 bet and a $10 call
    • Your call: match the bet or fold?
    • Pot odds: 1.5:1
    • Hand odds: 1.9:1 (or 0.35)
    • The pot odds do not outweigh the hand odds, so do not continue
computing hand odds vs pot odds1
Computing hand odds vs. pot odds
  • Consider the following hand progression:
  • Your hand: almost a flush (4 out of 5 cards of one suit)
    • Called a flush draw
  • On the flop: now a $30 pot, $10 bet and a $10 call
    • Your call: match the bet or fold?
    • Pot odds: 4:1
    • Hand odds: 1.9:1 (or 0.35)
    • The pot odds dooutweigh the hand odds, so do continue
more advanced texas hold em
More advanced Texas Hold’Em
  • There are other odds to consider:
    • Expected odds (what you expect other players in the game to bet on)
    • Your knowledge of the players
      • Both on how they bet in general
        • How often do they bluff, etc.
      • And any “things” that give away their hand
        • I.e. not keeping a “poker face”
    • Etc.
as an aside
As an aside?
  • What is the probably the worst pocket to be dealt in Texas Hold’em?
    • Alternatively, what is the worst initial two cards to be dealt in any poker game?
  • 2 and 7 of different suits
    • They are low cards, different suits, and you can’t do anything with them (they are just out of straight range)
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