1

Introduction to Discrete Probability

Epp, section 6.x

CS 202

Aaron Bloomfield

2

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

3

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

S

EEp )(

4

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

5

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

66

PokerPoker

7

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)

8

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

9

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

10

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

51481

4

5

13

11

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

37442

4

1

12

3

4

1

13

12

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

13

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

549121

4

1

4

2

12

3

4

1

13

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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

1616

Back to theory againBack to theory again

17

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

)(1 EpEp

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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

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Probability of the union of two events

S

E1 E2

p(E1 U E2)

20

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

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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

22

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…

23

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

2424

BlackjackBlackjack

25

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

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Blackjack table

27

Blackjack probabilities

• Getting 21 on the first two cards is called a blackjack– Or a “natural 21”

• Assume there is only 1 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)

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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)

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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…

30

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

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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

OUTWEIGHS the 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

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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

33

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?

34

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

35

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…

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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

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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

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• 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

As seen ina casino

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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?

41

• 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.

As seen ina casino

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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…

63.53$30

3*40$2*16$3*20$3*11$1*1000$3*30$

43

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!

4444

RouletteRoulette

45

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

46

The Roulette table

47

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

48

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

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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?

50

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 1 quadrillion

• See http://en.wikipedia.org/wiki/Martingale_(roulette_system) for more info

5151

Monty Hall ParadoxMonty Hall Paradox

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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?

53

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

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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

5555

A bit more theoryA bit more theory

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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 probability is the product of the individual probabilities

57

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

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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

6060

Texas Hold’emTexas Hold’em

Reference:Reference:

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

61

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

62

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

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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 4th community 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

64

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…

65

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

66

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

67

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

68

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

69

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!

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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

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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

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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

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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

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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 do outweigh the hand odds, so do

continue

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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.

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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)