Introduction to Discrete Probability Epp section 6 x
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: – 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 • 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) • 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 four of a kind or full house 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 – 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: • 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 E 1 and E 2 be events in sample space S • Then p(E 1 U E 2) = p(E 1) + p(E 2) – p(E 1 ∩ E 2) • Consider a Venn diagram dart-board 18
Probability of the union of two events p(E 1 U E 2) S E 1 E 2 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 $146 M – Thus, one should only play if the jackpot is greater than $146 M • If you count in the other prizes, then you will “break even” if the jackpot is $121 M 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 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 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 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 • 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 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: Payout: 1/38 2/38 4/38 18/38 36 x 18 x 9 x 2 x 2 x 48
Roulette • It has been 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 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 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 62
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 4 th community card is shown • River: when the 5 th 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 63 • 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… 64
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 65
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. 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 66
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 67
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 68
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! 69
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 70
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 71
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 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 • 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 74
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. 75
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) 76
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