MATH/COMP 3808 Mathematical Analyses of Games of Chance Winter 2015 Jason Gao Lecture 16, Mar. 04 The Bonus Bet in Caribbean Poker Expected Value of the Bonus Bet A Hand Flush Full House Four of a Kind Straight Flush Royal Flush Hit frequency Return to Player House Edge B C BC Payback Probability 50 0.00197 0.0985 100 0.00144 0.144 500 0.00024 0.12 20000 0.000014 0.28 200000 0.0000015 0.3 0.0036655 0.9425 0.0575 Call or Fold Let p denote the probability that dealer's hand does not qualify, i.e., lower than (A,K)-high. We note that the player's expected value of the call option is at least 3 1 = 4 3. The player's expected value of the surrender option is always 1. Hence the call option is correct when 4 3 1, or 0.5. Qualify or Not Note that, without any information, = 43.681%. This partly explains why (A,K)-high is the minimum qualifying hand, instead of one pair. Once we see 6 cards (5 in the player's hand, and 1 upcard from the dealer), the actual value of p may vary considerably. Example: {AK234} vs. A We note that this hand can only win when dealer's hand does not qualify. To find the number of dealer's hands which are lower than (A,K)-high, we may consider the following cases. Case 1. The dealer's hand is of the form {A,x,y,u,v} where x,y,u,v are not in {A,K,2,3,4}. The number of such hands is equal to 17850. Example: {AK234} vs. A Case 2. The dealer's hand is of the form {A,x,y,u,v} such that exactly one of them is in {2,3,4}. The number of such hands is equal to 32088. Case 3. The dealer's hand is of the form {A,x,y,u,v} such that exactly two of them are in {2,3,4}. The number of such hands is equal to 12012. Example: {AK234} vs. A Case 4. The dealer's hand is of the form {A,2,3,4,u}, where u is not in {A,K,2,3,4,5}. The number of such hands is equal to 7 4 33 1 = 749. Hence the total number of dealer's nonqualifying hands is 62699, and the probability that dealer's hand does not qualify is 62699/C(46,4) 38.42% Fold or Not? {AK234} vs. A If the player surrenders, he loses $1. Now we consider the option of not surrendering. The dealer's hand may tie with the player's hand, and this happens with probability 34 1 /C(46,4) 0.05%. Fold or Not? {AK234} vs. A In all other cases the player loses $3, and the corresponding probability is 1 38.42% 0.05% = 61.53%. Hence the expected value of the call option is 0.38423 0.6153 = 1.46. Since the expected value of the non-surrender option is smaller than 1, the correct play is to surrender. Example: {AK234} vs. 4 It becomes more tedious (about 12 cases to analyze) to compute the number of nonqualify hands in this case. One can use a computer program to find such number to be 81670. Hence the probability that the dealer does not qualify is 81670/C(46,4)=0.5005. Hence it is (slightly) better to call (not fold) in this case. Fold or Not? Examples Player's Hand Dealer's Up Card Call EV AK234 A 1.46 Correct Action Fold AK234 4 0.9966 Call 33456 9 0.815 Call 33456 A 0.55 Call 33456 2 0.16 Call 22456 10 0.97 Call Fold or Not? Examples Player's Hand Dealer's Up Card Call EV AKJ32 A 0.968 Correct Action Call AKJ32 6 1.065 Fold AKJ83 6 1.04 Fold AKJ83 3 0.835 Call AKQ102 6 1.0001 Fold Fold or Call ? Ignore Dealer's Upcard Player's Hand Call EV AKJ82 1.00138 Correct Action Fold AKJ83 0.999456 Call Exact Analysis Is Computationally Hard An exact analysis involves 47C(52,5)C(46,4)2 1013 possibilities. Suppose a computer program handles 105 cases/second, then it will take more than 1000 days to go through all the cases in Caribbean Stud Poker. Even if using equivalent classes, it still takes at least 1000/12 41 days. Computer Programming vs. Mathematical Analysis The number of unordered selections of 2 distinct numbers from {1,2,...10} is C(10,2) = 10 9/2 = 45. The following simple program also counts all such unordered selections. Initialize N:=0. For from 1 to 9 do for j from +1 to 10 do N:=N+1 End A Maple Program Sum of Three Dice