Introduction to Discrete Probability. Terminology. Probability definition. 22c:19, section 6.x Hantao Zhang

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1 Introduction to Discrete Probability 22c:19, section 6.x Hantao Zhang 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: E p ( E) = S Where E is the set of desired events (outcomes) and 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 1

2 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 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) 6 2

3 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 7 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 Poker probability: flush What is the chance of getting a flush? That s all 5 cards of the same suit We must do ALL of the following: Pick the suit for the flush: C(4,1) Pick the 5 cards in that suit: C(13,5) As we must do all of these, we multiply the values out (via the product rule) This yields 13 4 = 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 as flush or royal flash, 9 then the number is really

4 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 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 11 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: Pickthefacefor 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

5 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 More on probabilities Let E be an event in a sample space S. probability of the complement of E is: p ( E) = 1 p( E) The 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 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 15 5

6 Probability of the union of two events p(e1 U E2) S E 1 E 2 16 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) orp(5 n) =p(2 n) +p(5 n) -p(10 n) = 50/ /100 10/100 =3/5 17 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 18 6

7 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 19 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 20 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 21 7

8 Blackjack table 22 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: FirstcardisanA, 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 Blackjack probabilities Another way to get 1 in 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 (0.0483) 24 8

9 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 25 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 26 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 ablackjack 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 awards OUTWEIGHS the benefit of the single deck! You cannot win money on a 6:5 blackjack table that uses 1 deck Remember, the house always wins 27 9

10 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 Bustonadrawona18 4 or above will bust: that s 10 (of 13) cards that will bust 10/13 = probability bilit to bust Bustonadrawona17 5 or above will bust: 9/13 = probability to bust Bustonadrawona16 6 or above will bust: 8/13 = probability to bust Bustonadrawona15 7 or above will bust: 7/13 = probability to bust Bustonadrawona14 8 or above will bust: 6/13 = probability to bust 28 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 29 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 30 10

11 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 31 The Roulette table 32 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/

12 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 Payout: 36x 18x 9x 2x 2x 2x 2x 34 Roulette It has been proven that no advantageous strategies exist Including: Learning thewheel 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? 35 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*2 50 = \$1,125,899,906,842,624 to win this way! That s 1 quadrillion See for more info 36 12

13 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 37 Huh? What s behind door number three? Player has equal chance to choose one of the doors. There are three cases. 38 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 39 13

14 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 1minusthe probability of it never happening Or the complement of it never happening 40 Probability vs. odds Consider an event that has a 1 in 3 chance of happening Probability is 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, your 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 = :1 odds means that you will lose as many times as you win 41 14

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