Introduction to Discrete
Probability
Rosen, section 5.1
CS/APMA 202
Aaron Bloomfield
1
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
2
Probability definition
The probability of an event occurring is:
p( E ) 



E
S
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 3
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
4
Poker
5
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)
6
Poker probability: royal flush
What is the chance of
getting 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
7
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
8
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 13  4   5148
5 1

 
Possibilities for 5 cards: C(52,5) = 2,598,960
Probability = 5148/2,598,960 = 0.00198

Or about 1 in 505
9
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)
13  4 12  4 
This yields       3744
 1  3  1  2 
Possibilities for 5 cards: C(52,5) = 2,598,960
Probability = 3744/2,598,960 = 0.00144

Or about 1 in 694
10
Inclusion-exclusion principle
The possible poker hands are (in increasing order):

Nothing
One pair

Two pair








Three of a kind
Straight
Flush
Full house
Four of a kind
Straight flush
Royal flush
cannot include two pair, three of a kind,
four of a kind, or full house
cannot include three of a kind, four of a kind, or
full house
cannot include four of a kind or full house
cannot include straight flush or royal flush
cannot include straight flush or royal flush
cannot include royal flush
11
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)
13  4 12  4  4 
This yields        54912
 1  3  2  1  1 
Possibilities for 5 cards: C(52,5) = 2,598,960
Probability = 54,912/2,598,960 = 0.0211

Or about 1 in 47
12
Poker hand odds
The possible poker hands are (in increasing
order):










Nothing
One pair
Two pair
Three of a kind
Straight
Flush
Full house
Four of a kind
Straight flush
Royal flush
1,302,540
1,098,240
123,552
54,912
10,200
5,140
3,744
624
36
4
0.5012
0.4226
0.0475
0.0211
0.00392
0.00197
0.00144
0.000240
0.0000139
0.00000154
13
A solution to commenting your
code

The commentator:
http://www.cenqua.com/commentator/
14
Back to theory again
16
More on probabilities
Let E be an event in a sample space S. The
probability of the complement of E is:

p E  1  p( E )

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
17
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
18
Probability of the union of two
events
p(E1 U E2)
S
E1
E2
19
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
20
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
21
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 > $12.3 million


As, on the “average” you will only make money that
way
Of course, “average” will require trillions of lotto
plays…
22
Lots of piercings…
This may be a bit disturbing…
23
Blackjack
24
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
25
Blackjack table
26
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
27
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 = 20.72
28
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…
29
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


You cannot win money on a 6:5 blackjack table that uses 1 deck
Remember, the house always wins
30
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
31
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?
32
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
33
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…
34
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
35
As seen in
a casino
This wheel
spun if:



is
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
36
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?
37
As seen in
a 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.
38
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
$30 * 3  $1000 *1  $11* 3  $20 * 3  $16 * 2  $40 * 3  
 $53.63
30
So use the formula:




p(winning) * payout ≥ investment
0.0473 * $53.63 ≥ $1
$2.54 ≥ $1
Still not there yet…
39
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!
40
Quick survey

a)
b)
c)
d)
I felt I understood Blackjack probability…
Very well
With some review, I’ll be good
Not really
Not at all
41
Quick survey

a)
b)
c)
d)
If I was going to spend money gambling,
would I choose Blackjack?
Definitely – a way to make money
Perhaps
Probably not
Definitely not – it’s a way to lose money
42
Today’s dose of demotivators
43
Roulette
44
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
45
The Roulette table
46
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
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
Payout:
36x
18x
9x
2x
2x
2x
2x
48
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

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
49
Quick survey

a)
b)
c)
d)
I felt I understood Roulette probability…
Very well
With some review, I’ll be good
Not really
Not at all
50
Quick survey

a)
b)
c)
d)
If I was going to spend money gambling,
would I choose Roulette?
Definitely – a way to make money
Perhaps
Probably not
Definitely not – it’s a way to lose money
51
Monty Hall Paradox
52
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
55
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
56
A bit more theory
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 twice
in a row?
Cases:




Event happening neither time:
Event happening first time:
Event happening second time:
Event happening both times:
1/6 * 1/6 = 1/36
1/6 * 5/6 = 5/36
5/6 * 1/6 = 5/36
5/6 * 5/6 = 25/36
For an event to happen twice, the probability is
the product of the individual probabilities
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
What’s the chance of that event happening at least
once?
Cases:




Event happening neither time:
Event happening first time:
Event happening second time:
Event happening both times:
1/6 * 1/6 = 1/36
1/6 * 5/6 = 5/36
5/6 * 1/6 = 5/36
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
59
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

I think I presented this wrong last time…
60
More demotivators
61
Texas Hold’em
Reference:
http://teamfu.freeshell.org/poker_odds.html
62
Texas Hold’em
The most popular poker variant today
Every player starts with two face down cards
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Called “hole” or “pocket” cards
Hence the term “ace in the hole”
Five cards are placed in the center of the table
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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
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Your two hole cards and the 5 common cards
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Texas Hold’em
Hand progression
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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”
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Bets are placed
The next community card is turned over (or dealt)
Called the “turn”
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Bets are placed
The last community card is turned over (or dealt)
Called the “river”
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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
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Odds of a Texas Hold’em hand
Pick any poker hand
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We’ll choose a royal flush
There are 4/2,598,960 possibilities
Chance of getting that in a Texas Hold’em game:
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Choose your royal flush: C(4,1)
Choose the remaining two cards: C(47,2)
Result is 4324 possibilities
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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 Hold’em probability that way,
though…
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An example of a hand using
Texas 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
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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
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Any other heart
Thus, you have 9 outs
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Continuing with that example
There are 47 unknown cards
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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?
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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
Out on turn only
Out on river only
Out on both turn and river
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38/47 * 37/46
9/47 * 38/46
38/47 * 9/46
9/47 * 8/46
= 0.65
= 0.16
= 0.16
= 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
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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
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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?
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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
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I.e. the chance that you will get your nut hand
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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
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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
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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?
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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
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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?
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We are assuming that your nut hand will win
A safe assumption for a flush, but not a tautology!
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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
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What if you have 2:1 hand odds (0.33 probability)?
0.33 * $40 ≥ $10
$13.33 > $10
Definitely worth it to continue!
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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
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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
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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
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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
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Pot odds are thus 2:1
Use the formula: p(winning) * payout ≥ investment
When is it worth it to continue?
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What if you have 3:1 hand odds (0.25 probability)?
0.25 * $30 ≥ $10
$7.50 < $10
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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
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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
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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)
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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
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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)
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Called a flush draw
On the flop: now a $30 pot, $10 bet and a $10
call
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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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Quick survey
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a)
b)
c)
d)
I felt I understood Texas Hold’em probability…
Very well
With some review, I’ll be good
Not really
Not at all
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Quick survey

a)
b)
c)
d)
If I was going to spend money gambling,
would I choose Texas Hold’em?
Definitely – a way to make money
Perhaps
Probably not
Definitely not – it’s a way to lose money
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For next semester…
Other games I should go over?
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Quick survey

a)
b)
c)
d)
I felt I understood the material in this slide set…
Very well
With some review, I’ll be good
Not really
Not at all
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Quick survey

a)
b)
c)
d)
The pace of the lecture for this slide set was…
Fast
About right
A little slow
Too slow
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Quick survey

a)
b)
c)
d)
How interesting was the material in this slide
set? Be honest!
Wow! That was SOOOOOO cool!
Somewhat interesting
Rather borting
Zzzzzzzzzzz
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Today’s demotivators
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