# The game of poker. Gambling and probability. Poker probability: royal flush. Poker probability: four of a kind

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1 The game of poker Gambling and probability CS231 Dianna Xu 1 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) 2 Poker probability: royal flush What is the chance of getting a royal flush? 10, J, Q, K, and A of the same suit There are only 4 possible royal flushes Cardinality for 5-cards: C(52,5) = 2,598,960 Poker probability: four of a kind What is the chance of getting 4 of a kind when dealt 5 cards? 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 is 13*48 = 624 Probability = 4/2,598,960 = Or about 1 in 650,000 3 Probability = 624/2,598,960 = Or 1 in Poker probability: flush What is the chance of a flush? 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) When we have an and, we multiply the values out (via the product rule): C(13, 5)* C(4, 1) = 5148 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 Poker probability: full house What is the chance of getting a full house? 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) C(13, 1)*C(4, 3)*C(12, 1)*C(4, 2) = 3744 Probability = 3744/2,598,960 = Or about 1 in

2 Inclusion-exclusion principle Poker probability: three of a kind 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 7 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) Pick the suits for the two other cards: C(4,1)*C(4,1) C(13,1)*C(4,3)*C(12,2)*C(4,1)*C(4,1) = Probability = 54,912/2,598,960 = Or about 1 in 47 8 Poker hand odds Probability axioms 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 Let E be an event in a sample space S. probability of the complement of E is: 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 Royal flush 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 ) 11 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 12 2

3 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? What if you gamble \$1 and have a 1/100 probability to win \$10? One way to determine if gambling is worth it: probability of winning * payout amount spent per play Expected Value The expected values of a process with outcomes of values which occur with probabilities is: 13 Expected values of gambling Gamble \$1, and have a ½ probability to win \$10 (10-1)*0.5+(-1)*0.5 = 4 Gamble \$1 and have a 1/100 probability to win \$10? (10-1)*0.01+(-1)*0.99 = -0.9 Another way to determine if gambling is worth it: 15 Expected value > 0 When is lotto worth it? Many older lotto games you have to choose 6 numbers from 1 to 48 Total possible choices are 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 16 Powerball lottery Blackjack Modern powerball lottery: 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 You need to do both -- apply the product rule, Total possibilities are 3,478,761* 42 = 146,107,962 While there are many sub prizes, the probability for the jackpot is about 1 in 146 million If you count in the other prizes, then you will break even if the jackpot is \$121M 17 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 18 3

6 The Roulette table The Roulette table Bets can be placed on: Probability: Bets can be placed on: Probability: Payout: A single number Two numbers Four numbers 1/38 2/38 4/38 A single number Two numbers Four numbers 1/38 2/38 4/38 36x 18x 9x All even numbers All odd numbers The first 18 nums All even numbers All odd numbers The first 18 nums Red numbers Red numbers Roulette 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? Martingale betting strategy Where you double your (outside) 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

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