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## Introduction to Discrete Probability

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Title: Introduction to Discrete Probability

1
Introduction to Discrete Probability
• Rosen, section 5.1
• CS/APMA 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

4
Dice probability
• What is the probability of getting snake-eyes
(two 1s) 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 1s
is the probability of getting a 1 AND the
probability of getting a second 1
• Thus, its 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

5
Poker
6
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)

7
Poker probability royal flush
• What is the chance ofgetting a royal flush?
• Thats 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

8
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 1348 624
• Probability 624/2,598,960 0.00024
• Or 1 in 4165

9
Poker probability flush
• What is the chance of getting a flush?
• Thats 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

10
Poker probability full house
• What is the chance of getting a full house?
• Thats 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

11
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

12
Poker probability three of a kind
• What is the chance of getting a three of a kind?
• Thats three cards of one face
• Cant 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 cant 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

13
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 5,140 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

14
A solution to commenting your code
• The commentator http//www.cenqua.com/commentator
/

15
End of lecture on 12 April 2005
16
Back 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
• The book calls this Theorem 1
• 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

18
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 n E2)
• Consider a Venn diagram dart-board

19
Probability of the union of two events
p(E1 U E2)
S
E1
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(2n) 50/100 (all the even numbers)
• p(5n) 20/100
• p(2n) and p(5n) p(10n) 10/100
• p(2n) or p(5n) p(2n) p(5n) - p(10n)
• 50/100 20/100 10/100
• 3/5

21
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 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 gt 12.3
million
• As, on the average you will only make money
that way
• Of course, average will require trillions of
lotto plays

23
Lots of piercings
• This may be a bit disturbing

24
Blackjack
25
Blackjack
• You are initially dealt two cards
• 10, J, Q and K all count as 10
• Ace is EITHER 1 or 11 (players 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

26
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
• (416)/(5251) 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
• (164)/(5251) 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, its 1 in 20.72

28
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 10 card C(16,1)
• Total possibilities is the product of the two
(64)
• Probability is 64/1,326 20.72

29
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
• (416)/(5252) 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
• (164)/(5252) 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, its 1 in 21.13 (vs. 20.72)
• In reality, most casinos use shoes of 6-8 decks
for this reason
• It slightly lowers the players chances of
getting a blackjack
• And prevents people from counting the cards

30
So always use a single deck, right?
• Most people think that a single-deck blackjack
table is better, as the players odds increase
• And you can try to count the cards
• But its usually not the case!
• Normal rules have a 32 payout for a blackjack
• If you bet 100, you get your 100 back plus 3/2
• Most single-deck tables have a 65 payout
• You get your 100 back plus 6/5 100 or 120
• 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
• You cannot win money on a 65 blackjack table
that uses 1 deck
• Remember, the house always wins

31
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 thats 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

32
• If the dealers 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,
its a push
• If the dealer does not have blackjack, you lose
proceeds normal
• Is this insurance worth it?

33
• 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
• Lets 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 2308 616 back for your 1,000
spent
• Or, using the formula p(winning) payout
investment
• 0.308 2 1
• 0.616 1
• Thus, its not worth it
• Buying insurance is considered a very poor option
for the player
• Hence, almost every casino offers it

34
Blackjack strategy
• These tables tell you the best move to do on each
hand
• The odds are still (slightly) in the houses
favor
• The house always wins

35
Why counting cards doesnt work well
• If you make two or three mistakes an hour, you
• 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

36
As seen ina casino
• This wheel is spun if
• You get a natural blackjack
• You place 1 on the spin the wheel square
• You lose the dollar either way
• You win the amount shown on the wheel

37
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?

38
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.

39
Back to the drawing board
• If you weight each spot by the amount it can
win, you get 1609 for 30 spots
• Thats 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

40
My theory
• I think the wheel is weighted so the 1,000 side
of the wheel is heavy and thus wont be chosen
• As the chooser is at the top
• But I never saw it spin, so I cant 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 Im probably still missing something here
• Remember that the house always wins!

41
Quick survey
• I felt I understood Blackjack probability
• Very well
• With some review, Ill be good
• Not really
• Not at all

42
Quick survey
• If I was going to spend money gambling, would I
choose Blackjack?
• Definitely a way to make money
• Perhaps
• Probably not
• Definitely not its a way to lose money

43
Todays dose of demotivators
44
Roulette
45
Roulette
• A wheel with 38 spots is spun
• Spots are numbered 1-36, 0, and 00
• European casinos dont 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
49
Roulette
• It has been proven that proven that no
• Including
• Learning the wheels biases
• Casinos regularly balance their Roulette wheels
• Martingale betting strategy
• Where you double your bet each time (thus making
up for all previous losses)
• It still wont work!
• You cant double your money forever
• It could easily take 50 times to achieve finally
win
• If you start with 1, then you must put in 1250
1,125,899,906,842,624 to win this way!
• See http//en.wikipedia.org/wiki/Martingale_(roule

50
Quick survey
• I felt I understood Roulette probability
• Very well
• With some review, Ill be good
• Not really
• Not at all

51
Quick survey
• If I was going to spend money gambling, would I
choose Roulette?
• Definitely a way to make money
• Perhaps
• Probably not
• Definitely not its a way to lose money

52
53
Whats 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
• 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
• Thus, your chance of winning doubles if you
change
• Huh?

54
End of lecture on 14 April 2005
• Although I want to start 1 slide back

55
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?
• 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

56
Whats 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 is still 1/100
• You didnt 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

57
A bit more theory
58
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
• Whats 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 1/6 5/6 5/36
• Event happening second time 5/6 1/6 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

59
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
• Whats the chance of that event happening at
least once?
• Cases
• Event happening neither time 1/6 1/6 1/36
• Event happening first time 1/6 5/6 5/36
• Event happening second time 5/6 1/6 5/36
• Event happening both times 5/6 5/6 25/36
• Its 35/36!
• For an event to happen at least once, its 1
minus the probability of it never happening
• Or 1 minus the compliment of it never happening

60
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 21 odds
• Meaning if you play it 3 (21) times, you will
lose 2 times for every 1 time you win
• This, if you have xy odds, you probability is
y/(xy)
• The y is usually 1, and the x is scaled
appropriately
• For example 2.21
• That probability is 1/(12.2) 1/3.2 0.313
• 11 odds means that you will lose as many times
as you win
• I think I presented this wrong last time

61
More demotivators
62
Texas Holdem
• Reference
• http//teamfu.freeshell.org/poker_odds.html

63
Texas Holdem
• 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

64
Texas Holdem
• 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

65
Texas Holdem 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
• 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

66
Odds of a Texas Holdem hand
• Pick any poker hand
• Well choose a royal flush
• There are 4/2,598,960 possibilities
• Chance of getting that in a Texas Holdem game
• Choose your royal flush C(4,1)
• Choose the remaining two cards C(47,2)
• Result is 4324 possibilities
• Or 1 in 601
• Or probability of 0.0017
• Well, not really, but close enough for this slide
set
• This is much more common than 1 in 649,740 for
stud poker!
• But nobody does Texas Holdem probability that
way, though

67
An example of a hand usingTexas Holdem
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 thats the best hand you can (realistically)
hope for with the cards you have
• There are 9 cards that will allow you to achieve
• Any other heart
• Thus, you have 9 outs

68
Continuing with that example
• There are 47 unknown cards
• The two unturned cards, the other players cards,
and the rest of the deck
• There are 9 outs (the other 9 hearts)
• Whats the chance you will get your flush?
• Rephrased whats the chance that you will get an
out on at least one of the remaining cards?
• For an event to happen at least once, its 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 1.91

69
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 players cards,
and the rest of the deck
• There are still 9 outs (the other 9 hearts)
• Whats the chance you will get your flush?
• 9/46 0.20
• Or 1 in 5.1
• Or 4.11
• The odds have significantly decreased!
• These odds are called the hand odds
• I.e. the chance that you will get your nut hand

70
Hand odds vs. pot odds
• So far weve 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 well 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
is not considered as part of the odds)
• Or 31
• When is it worth it to continue?
• What if you have 31 hand odds (0.25
probability)?
• What if you have 21 hand odds (0.33
probability)?
• What if you have 11 hand odds (0.50
probability)?
• Note that we did not consider the probabilities
before the flop

71
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 31 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 21 hand odds (0.33
probability)?
• 0.33 40 10
• 13.33 gt 10
• Definitely worth it to continue!
• What if you have 11 hand odds (0.50
probability)?
• 0.5 40 10
• 20 gt 10
• Definitely worth it to continue!

72
Pot odds
• Pot odds is the ratio of the amount in the pot to
the amount you have to call
• In other words, we dont 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 31
• As there was 30 in the pot, and the call was 10
• Even though you invested some money previously

73
Another take on pot odds
• Assume the pot is 100, and the call is 10
• Thus, the pot odds are 10010 or 101
• 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 101
• If you have better odds, youll make money in the
long run
• If you have worse odds, youll lose money in the
long run

74
Hand odds vs. pot odds
• Pot is now 20, investment is 10
• Pot odds are thus 21
• Use the formula p(winning) payout investment
• When is it worth it to continue?
• What if you have 31 hand odds (0.25
probability)?
• 0.25 30 10
• 7.50 lt 10
• What if you have 21 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 11 hand odds (0.50
probability)?
• 0.5 30 10
• 15 gt 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

75
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 1.51
• Hand odds 1.91 (or 0.35)
• The pot odds do not outweigh the hand odds, so do
not continue

76
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 41
• Hand odds 1.91 (or 0.35)
• The pot odds do outweigh the hand odds, so do
continue

77
Quick survey
• I felt I understood Texas Holdem probability
• Very well
• With some review, Ill be good
• Not really
• Not at all

78
Quick survey
• If I was going to spend money gambling, would I
choose Texas Holdem?
• Definitely a way to make money
• Perhaps
• Probably not
• Definitely not its a way to lose money

79
For next semester
• Other games I should go over?

80
Quick survey
• I felt I understood the material in this slide
set
• Very well
• With some review, Ill be good
• Not really
• Not at all

81
Quick survey
• The pace of the lecture for this slide set was
• Fast
• A little slow
• Too slow

82
Quick survey
• How interesting was the material in this slide
set? Be honest!
• Wow! That was SOOOOOO cool!
• Somewhat interesting
• Rather borting
• Zzzzzzzzzzz

83
Todays demotivators
84
End of lecture on 19 April 2005