Introduction to Discrete Probability

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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 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

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
100, or 150 additional
Most single-deck tables have a 65 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 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
Buying (blackjack) insurance

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
your insurance bet, but your original bet
proceeds normal
Is this insurance worth it?

33
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
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
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

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
advantageous strategies exist
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!
Thats 1 quadrillion
See http//en.wikipedia.org/wiki/Martingale_(roule
tte_system) for more info

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
Monty Hall Paradox
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
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?

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?
Does that chance change if I tell you more info
about whats 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

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 before your switch, your chance of being
right is still 1/100
Just because you know more info about the other
doors doesnt change your chances
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
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

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
your flush
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
Your investment is 10
Your pot odds are 3010 (not 4010, as your call
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