1. Introduction to Discrete Probability
Epp, section 6.x
CS 202
Aaron Bloomfield

2. Terminology Experiment Sample space Event
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. Probability is always a value between 0 and 1
Something with a probability of 0 will never occur
Something with a probability of 1 will always occur
You cannot have a probability outside this range!
Note that when somebody says it has a “100% probability)
That means it has a probability of 1

5. Dice probability
What is the probability of getting “snake-eyes” (two 1’s) on two six-sided dice?
Probability of getting a 1 on a 6-sided die is 1/6
Via product rule, probability of getting two 1’s is the probability of getting a 1 AND the probability of getting a second 1
Thus, it’s 1/6 * 1/6 = 1/36
What is the probability of getting a 7 by rolling two dice?
There are six combinations that can yield 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
Thus, |E| = 6, |S| = 36, P(E) = 6/36 = 1/6

6. Poker

7. The game of poker You are given 5 cards (this is 5-card stud poker)
The goal is to obtain the best hand you can
The possible poker hands are (in increasing order):
No pair
One pair (two cards of the same face)
Two pair (two sets of two cards of the same face)
Three of a kind (three cards of the same face)
Straight (all five cards sequentially – ace is either high or low)
Flush (all five cards of the same suit)
Full house (a three of a kind of one face and a pair of another face)
Four of a kind (four cards of the same face)
Straight flush (both a straight and a flush)
Royal flush (a straight flush that is 10, J, K, Q, A)

8. Poker probability: royal flush
What is the chance 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 =
Or about 1 in 650,000

9. Poker probability: four of a kind
What is the chance of getting 4 of a kind when dealt 5 cards?
Possibilities for 5 cards: C(52,5) = 2,598,960
Possible hands that have four of a kind:
There are 13 possible four of a kind hands
The fifth card can be any of the remaining 48 cards
Thus, total possibilities is 13*48 = 624
Probability = 624/2,598,960 =
Or 1 in 4165

10. Poker probability: flush
What is the chance of getting a flush?
That’s all 5 cards of the same suit
We must do ALL of the following:
Pick the suit for the flush: C(4,1)
Pick the 5 cards in that suit: C(13,5)
As we must do all of these, we multiply the values out (via the product rule)
This yields
Possibilities for 5 cards: C(52,5) = 2,598,960
Probability = 5148/2,598,960 =
Or about 1 in 505
Note that if you don’t count straight flushes (and thus royal flushes) as a “flush”, then the number is really 5108

11. Poker probability: full house
What is the chance of getting a full house?
That’s three cards of one face and two of another face
We must do ALL of the following:
Pick the face for the three of a kind: C(13,1)
Pick the 3 of the 4 cards to be used: C(4,3)
Pick the face for the pair: C(12,1)
Pick the 2 of the 4 cards of the pair: C(4,2)
As we must do all of these, we multiply the values out (via the product rule)
This yields
Possibilities for 5 cards: C(52,5) = 2,598,960
Probability = 3744/2,598,960 =
Or about 1 in 694

12. Inclusion-exclusion principle
The possible poker hands are (in increasing order):
Nothing
One pair cannot include two pair, three of a kind, four of a kind, or full house
Two pair cannot include three of a kind, four of a kind, or full house
Three of a kind cannot include four of a kind or full house
Straight cannot include straight flush or royal flush
Flush cannot include straight flush or royal flush
Full house
Four of a kind
Straight flush cannot include royal flush
Royal flush

13. Poker probability: three of a kind
What is the chance of getting a three of a kind?
That’s three cards of one face
Can’t include a full house or four of a kind
We must do ALL of the following:
Pick the face for the three of a kind: C(13,1)
Pick the 3 of the 4 cards to be used: C(4,3)
Pick the two other cards’ face values: C(12,2)
We can’t pick two cards of the same face!
Pick the suits for the two other cards: C(4,1)*C(4,1)
As we must do all of these, we multiply the values out (via the product rule)
This yields
Possibilities for 5 cards: C(52,5) = 2,598,960
Probability = 54,912/2,598,960 =
Or about 1 in 47

14. Poker hand odds The possible poker hands are (in increasing order):
Nothing 1,302,
One pair 1,098,
Two pair 123,
Three of a kind 54,
Straight 10,
Flush 5,
Full house 3,
Four of a kind
Straight flush
Royal flush

15. End of lecture on 28 February 2007
And class was optional on Friday, 2 March 2007

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:
Recall the probability for getting a royal flush is
The probability of not getting a royal flush is or
Recall the probability for getting a four of a kind is
The probability of not getting a four of a kind is or

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 ∩ 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(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 – 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
For $100 spent, you win (on average) $10
Average win is $0.10 (or $10 * 1/100) for every $1 spent
One way to determine if gambling is worth it:
probability of winning * payout ≥ amount spent
Or p(winning) * payout ≥ investment
Of course, this is a statistical measure

22. When is lotto worth it?
Many older lotto games you have to choose 6 numbers from 1 to 48
Total possible choices is C(48,6) = 12,271,512
Total possible winning numbers is C(6,6) = 1
Probability of winning is
Or 1 in 12.3 million
If you invest $1 per ticket, it is only statistically worth it if the payout is > $12.3 million
As, on the “average” you will only make money that way
Of course, “average” will require trillions of lotto plays…

23. Powerball lottery Modern powerball lottery is a bit different
Source:
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

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 (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

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
= (4*16)/(52*51) = (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) = (or about 1 in 41)
Total chance of getting a blackjack is the sum of the two:
p = , or about 1 in 21
How appropriate!
More specifically, it’s 1 in (0.048)

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 = 1 in (0.048)

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
= (4*16)/(52*52) = (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) = (or about 1 in 42)
Total chance of getting a blackjack is the sum:
p = , or about 1 in 21
More specifically, it’s 1 in (vs )
In reality, most casinos use “shoes” of 6-8 decks for this reason
It slightly lowers the player’s chances of getting a blackjack
And prevents people from counting the cards…

30. Counting cards and Continuous Shuffling Machines (CSMs)
Counting cards means keeping track of which cards have been dealt, and how that modifies the chances
There are “easy” ways to do this – count all aces and 10-cards instead of all cards
Yet another way for casinos to get the upper hand
It prevents people from counting the “shoes” of 6-8 decks of cards
After cards are discarded, they are added to the continuous shuffling machine
Many blackjack players refuse to play at a casino with one
So they aren’t used as much as casinos would like

31. So always use a single deck, right?
Most people think that a single-deck blackjack table is better, as the player’s odds increase
And you can try to count the cards
But it’s usually not the case!
Normal rules have a 3:2 payout for a blackjack
If you bet $100, you get your $100 back plus 3/2 * $100, or $150 additional
Most single-deck tables have a 6:5 payout
You get your $100 back plus 6/5 * $100 or $120 additional
This lowered benefit of being able to count the cards OUTWEIGHS the benefit of the single deck!
And thus the benefit of counting the cards
Even with counting cards
You cannot win money on a 6:5 blackjack table that uses 1 deck
Remember, the house always wins

32. 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 = probability to bust
Bust on a draw on a 17
5 or above will bust: 9/13 = probability to bust
Bust on a draw on a 16
6 or above will bust: 8/13 = probability to bust
Bust on a draw on a 15
7 or above will bust: 7/13 = probability to bust
Bust on a draw on a 14
8 or above will bust: 6/13 = probability to bust

33. Buying (blackjack) insurance
If the dealer’s visible card is an Ace, the player can buy insurance against the dealer having a blackjack
There are then two bets going: the original bet and the insurance bet
If the dealer has blackjack, you lose your original bet, but your insurance bet pays 2-to-1
So you get twice what you paid in insurance back
Note that if the player also has a blackjack, it’s a “push”
If the dealer does not have blackjack, you lose your insurance bet, but your original bet proceeds normal
Is this insurance worth it?

34. Buying (blackjack) insurance
If the dealer shows an Ace, there is a 4/13 = probability that they have a blackjack
Assuming an infinite deck of cards
Any one of the “10” cards will cause a blackjack
If you bought insurance 1,000 times, it would be used 308 (on average) of those times
Let’s say you paid $1 each time for the insurance
The payout on each is 2-to-1, thus you get $2 back when you use your insurance
Thus, you get 2*308 = $616 back for your $1,000 spent
Or, using the formula p(winning) * payout ≥ investment
0.308 * $2 ≥ $1
0.616 ≥ $1
Thus, it’s not worth it
Buying insurance is considered a very poor option for the player
Hence, almost every casino offers it

35. Blackjack strategy
These tables tell you the best move to do on each hand
The odds are still (slightly) in the house’s favor
The house always wins…

36. End of lecture on 12 March 2007

37. 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

38. So why is Blackjack so popular?
Although the casino has the upper hand, the odds are much closer to 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

39. As seen in a casino This wheel is spun if:
You place $1 on the “spin the wheel” square
You get a natural blackjack
You lose the dollar either way
You win the amount shown on the wheel

40. 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 = , or 1 in 21.13
So use the formula:
p(winning) * payout ≥ investment
* $ ≥ $1
$4.90 ≥ $1
But the house always wins! So what happened?

41. 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.

42. Back to the drawing board
If you weight each “spot” by the amount it can win, you get $1609 for 30 “spots”
That’s an average of $53.63 per spot
So use the formula:
p(winning) * payout ≥ investment
* $53.63 ≥ $1
$2.54 ≥ $1
Still not there yet…

43. My theory
I think the wheel is weighted so the $1,000 side of the wheel is heavy and thus won’t be chosen
As the “chooser” is at the top
But I never saw it spin, so I can’t say for sure
Take the $1,000 out of the 30 spot discussion of the last slide
That leaves $609 for 29 spots
Or $21.00 per spot
So use the formula:
p(winning) * payout ≥ investment
* $21 ≥ $1
$ ≥ $1
And I’m probably still missing something here…
Remember that the house always wins!

44. Roulette

45. Roulette A wheel with 38 spots is spun
Spots are numbered 1-36, 0, and 00
European casinos don’t have the 00
A ball drops into one of the 38 spots
A bet is placed as to which spot or spots the ball will fall into
Money is then paid out if the ball lands in the spot(s) you bet upon

46. The Roulette table

47. The Roulette table Bets can be placed on: Probability: 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

48. The Roulette table Bets can be placed on: Probability: Payout:
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
Payout:
36x
18x 9x 2x

49. Roulette
It has been proven that proven that no advantageous strategies exist
Including:
Learning the wheel’s biases
Casino’s regularly balance their Roulette wheels
Using lasers (yes, lasers) to check the wheel’s spin
What casino will let you set up a laser inside to beat the house?

50. Roulette
It has been proven that proven that no advantageous strategies exist
Including:
Martingale betting strategy
Where you double your bet each time (thus making up for all previous losses)
It still won’t work!
You can’t double your money forever
It could easily take 50 times to achieve finally win
If you start with $1, then you must put in $1*250 = $1,125,899,906,842,624 to win this way!
That’s 1 quadrillion
See for more info

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 No!
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

54. 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:

55. A bit more theory

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 twice in a row?
Cases:
Event happening neither time: 1/6 * 1/6 = 1/36
Event happening first time: 5/6 * 1/6 = 5/36
Event happening second time: 1/6 * 5/6 = 5/36
Event happening both times: 5/6 * 5/6 = 25/36
For an event to happen twice, the probability is the product of the individual probabilities

57. An aside: probability of multiple events
Assume you have a 5/6 chance for an event to happen
Rolling a 1-5 on a die, for example
What’s the chance of that event happening at least once?
Cases:
Event happening neither time: 1/6 * 1/6 = 1/36
Event happening first time: 5/6 * 1/6 = 5/36
Event happening second time: 1/6 * 5/6 = 5/36
Event happening both times: 5/6 * 5/6 = 25/36
It’s 35/36!
For an event to happen at least once, it’s 1 minus the probability of it never happening
Or 1 minus the compliment of it never happening

58. 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

59. End of lecture on 16 March 2007

60. Reference: http://teamfu.freeshell.org/poker_odds.html
Texas Hold’em
Reference:

61. Texas Hold’em The most popular poker variant today
Every player starts with two face down cards
Called “hole” or “pocket” cards
Hence the term “ace in the hole”
Five cards are placed in the center of the table
These are common cards, shared by every player
Initially they are placed face down
The first 3 cards are then turned face up, then the fourth card, then the fifth card
You can bet between the card turns
You try to make the best 5-card hand of the seven cards available to you
Your two hole cards and the 5 common cards

62. Texas Hold’em Hand progression Note that anybody can fold at any time
Cards are dealt: 2 “hole” cards per player
5 community cards are dealt face down (how this is done varies)
Bets are placed based on your pocket cards
The first three community cards are turned over (or dealt)
Called the “flop”
Bets are placed
The next community card is turned over (or dealt)
Called the “turn”
The last community card is turned over (or dealt)
Called the “river”
Hands are then shown to determine who wins the pot

63. Texas Hold’em terminology
Pocket: your two face-down cards
Pocket pair: when you have a pair in your pocket
Flop: when the initial 3 community cards are shown
Turn: when the 4th community card is shown
River: when the 5th community card is shown
Nuts (or nut hand): the best possible hand that you can hope for with the cards you have and the not-yet-shown cards
Outs: the number of cards you need to achieve your nut hand
Pot: the money in the center that is being bet upon
Fold: when you stop betting on the current hand
Call: when you match the current bet

64. Odds of a Texas Hold’em hand
Pick any poker hand
We’ll choose a royal flush
There are only 4 possibilities (1 of each suit)
There are 7 cards dealt
Total of C(52,7) = 133,784,560 possibilities
Chance of getting that in a Texas Hold’em game:
Choose the 5 cards of your royal flush: C(4,1)
Choose the remaining two cards: C(47,2)
Product rule: multiply them together
Result is 4324 (of 133,784,560) possibilities
Or 1 in 30,940
Or probability of 0.000,032
This is much more common than 1 in 649,740 for stud poker!
But nobody does Texas Hold’em probability that way, though…

65. An example of a hand 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

66. Continuing with that example
There are 47 unknown cards
The two unturned cards, the other player’s cards, and the rest of the deck
There are 9 outs (the other 9 hearts)
What’s the chance you will get your flush?
Rephrased: what’s the chance that you will get an out on at least one of the remaining cards?
For an event to happen at least once, it’s 1 minus the probability of it never happening
Chances:
Out on neither turn nor river 38/47 * 37/46 = 0.65
Out on turn only 9/47 * 38/46 = 0.16
Out on river only 38/47 * 9/46 = 0.16
Out on both turn and river 9/47 * 8/46 = 0.03
All the chances add up to 1, as expected
Chance of getting at least 1 out is 1 minus the chance of not getting any outs
Or = 0.35
Or 1 in 2.9
Or 1.9:1

67. Continuing with that example
What if you miss your out on the turn
Then what is the chance you will hit the out on the river?
There are 46 unknown cards
The two unturned cards, the other player’s cards, and the rest of the deck
There are still 9 outs (the other 9 hearts)
What’s the chance you will get your flush?
9/46 = 0.20
Or 1 in 5.1
Or 4.1:1
The odds have significantly decreased!
These odds are called the hand odds
I.e. the chance that you will get your nut hand

68. Hand odds vs. pot odds
So far we’ve seen the odds of getting a given hand
Assume that you are playing with only one other person
If you win the pot, you get a payout of two times what you invested
As you each put in half the pot
This is called the pot odds
Well, almost – we’ll see more about pot odds in a bit
After the flop, assume that the pot has $20, the bet is $10, and thus the call is $10
Payout (if you match the bet and then win) is $40
Your investment is $10
Your pot odds are 30:10 (not 40:10, as your call is not considered as part of the odds)
Or 3:1
When is it worth it to continue?
What if you have 3:1 hand odds (0.25 probability)?
What if you have 2:1 hand odds (0.33 probability)?
What if you have 1:1 hand odds (0.50 probability)?
Note that we did not consider the probabilities before the flop

69. Hand odds vs. pot odds Pot payout is $40, investment is $10
Use the formula: p(winning) * payout ≥ investment
When is it worth it to continue?
We are assuming that your nut hand will win
A safe assumption for a flush, but not a tautology!
What if you have 3:1 hand odds (0.25 probability)?
0.25 * $40 ≥ $10
$10 = $10
If you pursue this hand, you will make as much as you lose
What if you have 2:1 hand odds (0.33 probability)?
0.33 * $40 ≥ $10
$13.33 > $10
Definitely worth it to continue!
What if you have 1:1 hand odds (0.50 probability)?
0.5 * $40 ≥ $10
$20 > $10

70. 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

71. 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

72. 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

73. 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

74. 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

75. 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.

76. 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)

77. End of lecture on 19 March 2007