Introduction to Discrete
Probability
Epp, section 6.x
CS 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
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
4
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
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 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
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 13*48 = 624
• Probability = 624/2,598,960 = 0.00024
– Or 1 in 4165
9
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
• Note that if you don’t count straight flushes (and thus royal flushes)
10
as a “flush”, then the number is really 5108
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 
      3744
 1  3  1  2 
•
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
– 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
12
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
13  4 12  4  4 
       54912
 1  3  2  1  1 
– Or about 1 in 47
13
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
5,108
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
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 )
• 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 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…
22
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
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 (0.048)
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 = 1 in 20.72 (0.048)
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
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
30
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
31
– Remember, the house always wins
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
32
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?
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
– 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
34
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…
35
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
37
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
38
As seen in
a casino
• This wheel
spun if:
is
– 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
39
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?
40
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.
41
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…
42
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!
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
– Using lasers (yes, lasers) to check the wheel’s
spin
• What casino will let you set up a laser inside to
beat the house?
49
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
50
Monty Hall Paradox
51
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?
52
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
53
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
54
A bit more theory
55
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
5/6 * 1/6 = 5/36
1/6 * 5/6 = 5/36
5/6 * 5/6 = 25/36
• For an event to happen twice, the probability is
the product of the individual probabilities
56
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
5/6 * 1/6 = 5/36
1/6 * 5/6 = 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
57
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
58
Texas Hold’em
Reference:
http://teamfu.freeshell.org/poker_odds.html
60
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
61
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
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• Call: when you match the current bet
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…
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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
– 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
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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
Out on turn only
Out on river only
Out on both turn and river
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
– 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
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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
– 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
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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?
– 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 72
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)
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