Introduction to discrete probability
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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


Poker

Poker


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 paircannot include two pair, three of a kind, four of a kind, or full house

    • Two paircannot include three of a kind, four of a kind, or full house

    • Three of a kindcannot include four of a kind or full house

    • Straightcannot include straight flush or royal flush

    • Flushcannot include straight flush or royal flush

    • Full house

    • Four of a kind

    • Straight flushcannot 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):

    • Nothing1,302,5400.5012

    • One pair1,098,2400.4226

    • Two pair123,5520.0475

    • Three of a kind54,9120.0211

    • Straight10,2000.00392

    • Flush5,1080.00197

    • Full house3,7440.00144

    • Four of a kind6240.000240

    • Straight flush360.0000139

    • Royal flush40.00000154


Back to theory again

Back to theory again


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 events1

Probability of the union of two events

p(E1 U E2)

S

E1

E2


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


Blackjack

Blackjack


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 table

Blackjack table


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!


Roulette

Roulette


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 table

The Roulette table


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


Monty hall paradox

Monty Hall Paradox


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


A bit more theory

A bit more theory


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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