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# DISCRETEMATHEMATICS MATH245

SDSU

GPA 3.72

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This 5 page Class Notes was uploaded by Burnice Ratke on Tuesday October 20, 2015. The Class Notes belongs to MATH245 at San Diego State University taught by Staff in Fall. Since its upload, it has received 13 views. For similar materials see /class/225273/math245-san-diego-state-university in Mathematics (M) at San Diego State University.

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Date Created: 10/20/15

Math 245 Discrete Mathematics Counting and Probability Gambling Like a Mathematican Lecture Notes 13 Peter Blomgren Department of Mathematics and Statistics San Diego State University San Diego CA 921827720 blomgrenterminus SDSU EDU http terminus SDSU EDU 51d lecturetexv 18 20061202 210828 blomgren Exp 5 Counting and Pxobabxhty Gambling Like a Mathematican e p 119 The Birthday Problem 1 of 2 Consider the Birthday problem The probability that n people have different birthdays is given by 736570 36571 365701707 365 n T T 365711 365quot 365 365 365 Pn The Probability for Different Birthdays Counting and Pxobabxhty Gambling Like a Mathematican e p 219 The Birthday Problem 2 of 2 This means that in a group of 23 or more people the probability that two persons have the same birthday is greater than 50 this is somewhat surprising In a class of 42 students the probability of 42 unique birthdays is 00860 86 Since the result above is somewhat surprising we can find some un lucky fool who has not taken this class and make quota friendly wager In a room of 23 people selected at random from the population you will win the bet at least two people have the same birthdayquot 5073 of the time If you bet with 139tov1 odds your average or expected payoff per dollar is PWn 2 130se 0 10146 so in the long run you make money Counting and Pxobabxhty Gambling Like a Mathematican e p 319 Expected Value Formally the expected value is defined as Definition Expected Value Suppose the possible outcomes of an experiment or random pro cess are real numbers a1 a2 an which occur with probabili ties 131 p2 13 The expected value of the process is 71 2am k1 Counting and Pxobabxhty Gambling Like a Mathematican e p 419 Expected Value Application California Lottery 1 of 2 The Super Lotto Plus draw consists of a Mega number in the range 1 27 and five numbers in the range 1 47 Hence there are 5 27 47gt 41416353 possible combinations The following combinations constitute winning tickets Combination Possibilities Oddsquot 5 of 5 Mega 1 1 in 41416353 5 of 5 26 1 in 1592936 4 of 5 Mega j 42 210 1 in 197221 4 of 5 3 4226 5460 1 in 7585 3 of 5 Mega i 422 8610 1 in 4810 2 of 5 Mega j 432 114800 1 in 361 3 of 5 Z 422 26 223860 1 in 185 1 of 5 Mega f 4f 559650 1 in 74 0 of 5 Mega g 452 850668 1 in 49 Ceunimg and Pxobabihby Gambling Like a Mabhemahcan e p 519 Expected Value Application California Lottery 2 of 2 In a draw 1837 11102004 the winning amounts where Combination Oddsquot Pk Value a0 ak Pk 5 of 5 Mega 1 in 41416353 39000000 0941657 5 of 5 1 in 1592936 35844 0022502 4 of 5 Mega 1 in 197221 1378 0006987 4 of 5 1 in 7585 104 0013711 3 of 5 Mega 1 in 4810 49 0010187 2 of 5 Mega 1 in 361 9 0024930 3 of 5 1 in 185 10 0054054 1 of 5 Mega 1 in 74 1 0013513 0 of 5 Mega 1 in 49 1 0020408 Expected Value 1107950 Hence in this particular draw the expected return of investment per dollar was 110 The jackpot starts at 7000000 in such a draw the expected value is approximately only 020 Ceunimg and Pxobabihby Gambling Like a Mabhemahcan e p 619 Application Mega Millions Even Worse Odds The Mega Millions draw consists of a Mega number in the range 1 46 and five numbers in the range 1 56 Hence there are 46 556 175711536 possible combinations The following combinations constitute winning tickets Combination Possibilities Oddsquot 5 of 5 Mega 1 1 in 175711536 5 of 5 45 1 in 3904701 4 or 5 Mega j 51 255 1 in 689065 4 of 5 j 51 45 11475 1 in 15313 3 of 5 Mega g 521 12750 1 in 13781 3 of 5 g 521 45 573750 1 in 306 2 of 5 Mega j 531 208250 1 in 844 1 of 5 Mega f 541 1249500 1 in 141 0 of 5 Mega g 551 2 349060 1 in 75 Ceunimg and Pxobabihby Gambling Like a Mabhemahcan e p 719 Expected Value Comments From a mathematical standpoint it makes sense to play the California lottery or any other state lottery eg powerball when the value ofthe jackpot approaches the number of combinations possible in the lottery Note that your chances of winning do not improve but the expected payoff is greater than the price of the ticket In all Las Vegas style gambling the expected payoff is less than oneetoeone so in the long run the house always wins In some games eg Blackjack the odds and therefore expected payoff change as the game progresses This is where counting cards keeping track of what cards have been played can help the player achieve an expected value greater than 1 Casinos combat this by shuffling as many as 8 decks of cards together and only playing half of the cards before the next reshuffle See Edward O Thorp Beat the Dealer A Winning Strategy for the Game of Twenty Onequot Vintage Revised edition April 12 1966 Ceunimg and Pxobabihby Gambling Like a Mabhemahcan e p 919 Poker SCard Draw 52card Deck There are 25987 960 possible poker hands In decreasing order of value the following are the hands of interest Hand Count Count Odds Probability Royal Flush 4 4 1 in 649740 1539110c6 Straight Flush 4 10 4 4 36 1 in 72193 1335210c5 FourofaKind 13 43 624 1 in 4165 24010104 Full House 13 126 3744 1 in 694 00014 Flush 40 4 36 4 4 5103 1 in 509 00020 Straight 10 45 4 36 4 4 10200 1 in 255 00039 ThreeofaKind 13 44 54912 1 in 47 00211 Two Pairs 123 2 44 123552 1 in 21 00475 One Pair 13gl13243 1093240 1 in 2 04226 Odds rounded to closest integer Caummg and Prababimy Gambling Like a Marhamaucan 4 p 919 Reference Poker Hands Royal Flush A4K4Q41410 or A94K94Q94194104 or AW KW QW JWIOW or AOKOQOJOIOO Straight Flush Sequence of five cards in either suit except A39K39 QJ 10 eg chQ rJ rlOJrQ FourofaKind EgAQA 7AJrAltgtX Fun House Threeaofaaakind and a pair eg A A AampKltgtKlt7 Hush All five cards of same suit but not in sequence Straight All five cards in sequence but not of the same suit Threeofakind Threeofakind and two other non39paired cards eg AQ39AW39A 39KO39IOW Two pairs Two different pairs and one other cardeg AQ39 A kyakltgt410lt7 Pair One pair and three other cards eg AQ39AW39KJrQO39JW C ounhng and Prababimy Gambling Like a Marhamaucan 4 p 1019 Texas Hold em Texas Hold39em is played 1 Prefop each player gets two pocket cards face down 2 Betting 3 The Flop 3 cards are put face39up on the table 4 Betting 6 Betting AAAAAA 01 N l l l The Turn one card is put face up on the table The River one card is put face up on the table 8 Betting Each player creates a poker hand out of five cards from hisher two pocket cards and the community cards on the table The strongest hand wins Here order and definitely psychology matters Caummg and Probabimy Gambling Like a Marhamaucan 4 p 1119 Texas Hold em Preflop 1 of 2 There are 1326 possible pocket card combinations The strongest preflop is a pair of aces of which there are 6 possibilities hence the probability for this event is 61326 00045 1 in 221 The probability for any specified pair is the same 61326 00045 1 in 221 and the probability for any some pair is 6 131326 00533 1 in 17 In a 3player game the probability that you have pocket aces and both the other players have pocket kings 3 3 3 36 17633 106 562 20 353 520 Here you are almost guaranteed a win how can you lose Caummg and Prababimy Gambling Like a Marhamaucan 4 p 1219 Texas Hold em Preflop 2 of 2 Hand Count Count Odds Probability Pair of Aces 3 6 1 in 221 00045 Some pair 133 78 1 in 17 00588 AceeKing suited 4 4 1 in 332 00030 AceeKing offesuitquot 43 4 1 in 111 00090 Two cards suited 4 123 312 1 in 4 02352 aha quotBig Slick This far it s pretty straight forward As long as you know that 67 123 78 and 1326 calculating the oddsprobabilities can be done quickly in your head with some practice Counting and Pxobabihty Gambling Like e Mathematican e p 1319 Texas Hold em The Flop 1 of 3 The op is something of a main event There are 22100 threeecard combinations that can op 5272 3 From the point of view of the TVeaudience there are possible ops in an nvplayer game since 211 cards are already known n 2 3 4 5 6 7 ops 17296 15180 13244 11480 9880 8436 From the point of view of a player holding two cards from the pres op there are now 530 197 600 possible ops From now on unless otherwise specified we take the stance of one player Counting and Pxobability Gambling Like e Mathematican e p 17119 Texas Hold em The Flop 2 of 3 You the player hold AQKQ from the pre op Flop Count Count Odds Probability Hand 2410 1 1 1 in 19600 51020 1075 Royal Flush AJrAUeAO 1 1 1 in 19600 51020 10 5 Foureofeaekind K KUeKO 1 1 1 in 19600 51020 10 5 Foureofeaekind AeAeK mi 9 1 in 2178 4591810 4 Full House AeK AeKeK mi 9 1 in 2178 4591810 4 Full House 19A A A non AK g 414 132 1 in 148 00067 ThreeeofeaeKind K K non AK g 414 132 1 in 148 00067 ThreeeofeaeKind Any 3 a 13 165 1 in 119 00084 Flush A K non AK mi 414 39s 1 in 49 00202 2 Pairs Any 2 a 704 121 319 2145 1 in 9 01094 Flush Draw1 No a A or K 3 5456 1 in 5 02784 Mostly Nothing 1 lncludes pair of aces and logical xor pair of kings 2 lncludes 27 A K Q J 10 straights and an untold number of straightedraws Counting and Pxobabihty Gambling Like e Mathematican e p 1519 Texas Hold em The Flop 3 of 3 You the player hold ngK from the pre op Flop Count Count Odds Probability Hand 3 Aces 3 4 1 in 4900 20408 1074 Full House A K1 KeAeA mg 12 1 in 1633 6123710 4 Full House K A2 KeKeother 48 48 1 in 408 00024 FoureofeaeKind K non A pair 111 132 1 in 148 00068 Full House Kother Any 3 a 132 220 1 in 89 00112 Flush Draw K non pair 384 1 in 51 00196 ThreeeofeaeKind Pair non K 122lt lt 1584 1 in 12 00808 2 rgtar39rs8 Any 2 a 122N319 2574 1 in 8 01313 Flush Draw4 3 nonepair noneK 132 14080 1 in 14 07184 PaireofeK 1 Loses to a player holding the remaining ace foureofeaekind 2 Loses to pocket aces foureofeaekind 3 Loses to threeeofeaekind 4 Long shot results in flush 1 in 24 00416 Counting and Pxobability Gambling Like e Mathematican e p 1619 Texas Hold em The Turn amp River 1 of 2 Texas Hold em The Turn amp River 2 of 2 You hold AQ39AO and the op is KchQQ39JO You hold AltgtQltgt and the op is 704024 Final Hand Count Count Odds Probability Comment Final Hand Count Count Odds Probability Note A AltgtA4Alt7 1 1 1 m 1001 92507 104 You win 0th g 3618 370 1 in 3 03497 AltAltKltKltK i 3 1 in 360 00020 A A A i 3 1 in 360 00028 AAAKK f f 0 1 m 100 00056 Q Q Q i 3 1 in 360 00020 AAAKK or go or 11 f f 10 1 m 00 00107 A A Q Q mi 9 1 in 120 00003 440011 pair f 21 40 1 m 22 00444 4477 mi 9 1 in 120 00003 1 444 814 72 1 m 15 00000 Q Q 7 7 mi 9 1 in 120 00003 1 AltAltKltK 613 114 1 in 9 01055 AA or go 814 144 1 in 0 01332 A K Q J lo 213 E 178 1 Iquot 6 01647 1 When the second pair comes completely from the board you are vulnerable since a 44 211 440 1 m 2 04144 single 7 in an opponents hand gives himher a three of a kind Note that two pairs is quite vulnerable since it loses to three39of39a kind With a pair on the table there is a 44 425 89990 00899 probability that a particular opponent has at three or fourofalltind Coummg and Pxobabxhty Gambhng kae a Mabhemahcan a p 1719 Coummg and Pxobabxhty Gambhng kae a Mabhemahcan a p 1919 What Are the Odds of Winning So far we have calculated the probabilities that you can form certain hands The real question is whether you will win or lose At every stage of the game you can in a similar way compute the probabilities that your opponents have stronger hands than you After the op there are 427 17 081 twocard combinations pocket cards your opponents may be holding If your hand is vulnerable to single cards then you want to force other opponents out of the game by betting aggressively this will reduce your risk of losing to a dumb lucllt handquot In the long run understanding the relative strengths of the hands and being able to quickly in your head compute rough estimates of the probabilities on the yquot should give you an advantage over the average player Just remember that in any given round of play dumb luck beats Skill Never bet more than you can afford to lose Coummg and Pxobabxhty Gambhng kae a Mabhemahcan a p 1919

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