Introduction to Discrete Probability

- Rosen, section 5.1
- CS/APMA 202
- Aaron Bloomfield

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

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

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

Poker

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)

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

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

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

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

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

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

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

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

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

Probability of the union of two events

p(E1 U E2)

S

E1

E2

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

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

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

Blackjack

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

Blackjack table

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

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

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

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

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

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?

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

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

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

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

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?

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.

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

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!

Roulette

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

The Roulette table

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

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

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

Monty Hall Paradox

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?

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

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

A bit more theory

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

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

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

Texas Holdem

- Reference
- http//teamfu.freeshell.org/poker_odds.html

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

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

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

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

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

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

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

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

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!

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

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

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

Computing hand odds vs. pot odds

- Consider the following hand progression
- Your hand almost a flush (4 out of 5 cards of

one suit) - Called a flush draw
- Perhaps because one more draw can make it a flush
- On the flop 5 pot, 10 bet and a 10 call
- Your call match the bet or fold?
- Pot odds 1.51
- Hand odds 1.91 (or 0.35)
- The pot odds do not outweigh the hand odds, so do

not continue

Computing hand odds vs. pot odds

- Consider the following hand progression
- Your hand almost a flush (4 out of 5 cards of

one suit) - Called a flush draw
- On the flop now a 30 pot, 10 bet and a 10

call - Your call match the bet or fold?
- Pot odds 41
- Hand odds 1.91 (or 0.35)
- The pot odds do outweigh the hand odds, so do

continue