Stat 35b: Introduction to Probability with Applications to Poker Outline for the day: Project A and teams. P(4 of a kind), etc. Expected value and pot odds, continued Yang / Kravchenko Violette / Elezra example. Reminders: HW2 is due Fri Jan 31. Read up through chapter 4. Luck vs. skill. pp 71-79. Any thoughts? Project A is due by email by Feb 6, 8pm. You may email me for your teammates’ email addresses.   u    u 

Teams: team a HUANG, RIHAN. ZHANG, HO KWAN. MCELROY, DAVID. team b LI, JESSE. THOMA, TREVOR. DITTMAR, MICHAEL. team c WANG, ERIC. LU, SIAO. GU, JIAQI. team d CHIANG, NICOLE. ZHANG, TONY. ALCORN, KIMBERLY. team e CHEN, ZHAOYUAN. LI, GAN. REITZ, CLAIRE. team f BARBER, THOMAS. MENG, QINGYI. SUEZAKI, PATRICK. team g LAM, SUI YUEN. ZHUANG, YUAN. LAHOZ GONZALEZ, LAURA. team h XIE, ZITONG. GU, YINGQIAN. LUAN, PEIYAO. team i CHEN, YIMING. TUZMEN, MEHMET. WANG, ZHAOXIN. team j POWER, JUSTIN. ALEXANDER, AREN. MAULEON, KIMBERLY. team k KIM, SO YOUNG. CHEN, XINYUAN. DUTTA, ANJALI. team l ZHOU, MENGRUI. YU, BRANDON. SANGHVI, ANAV. team m GAO, YANG. LA, CASEY. TSAI, PEI-CHEN. team n LIU, YIHAO. LI, RUOYU. SHI, BO. team o ZHOU, CAROL. YUN, XINYAO. DUONG, MINDY. team p MAVRODIEV, TEODOR. WONG, LAP KAN. YUZAWA, KATSUHIRO. team q LEE, FIONA. YANG, YIK. HUSSEIN, MARYAM. MOUTOUX, ALEX. team r LI, JIANJUN. MUSTAFA, JAMAL. YANG, YI-LONG. HSUEH, JESSICA. Just submit one email per team.

Winning code last time. legendary = function(numattable1, crds1, board1, round1, currentbet, mychips1, pot1, roundbets, blinds1, chips1, ind1, dealer1, tablesleft){ ## all in if pair of 9s or better, ## or same suit with the lower card is 10 or above, ## or different suit with lower card is Jack or above, ## or you have less than 4 times big blind and at least one card is Jack or higher. a1 = 0 if((crds1[1,1] == crds1[2,1]) && (crds1[2,1] > 8.5)) a1 = mychips1 if((crds1[2,1] == crds1[2,2]) && (crds1[2,1] > 9.5)) a1 = mychips1 if((crds1[2,1] > 10.5)) a1 = mychips1 if((mychips1 < 4 * blinds1) && (crds1[1,1] > 10.5)) a1 = mychips1 a1 } ## end of legendary

Suppose you’re all in next hand, no matter what cards you get. P(eventually make 4-of-a-kind)? [including case where all 4 are on board] Trick: just forget card order, and consider all collections of 7 cards. Out of choose(52,7) different combinations, each equally likely, how many of them involve 4-of-a-kind? 13 choices for the 4-of-a-kind. For each such choice, there are choose(48,3) possibilities for the other 3 cards. So, P(4-of-a-kind) = 13 * choose(48,3) / choose(52,7) ~ 0.168%, or 1 in 595. P(flop 4-of-a-kind) = 13*48 / choose(52,5) = 0.024% = 1 in 4165. P(flop 4-of-a-kind | pocket pair)? No matter which pocket pair you have, there are choose(50,3) possible flops, each equally likely, and how many of them give you 4-of-a-kind? 48. (e.g. if you have 7 7, then need to flop 7u 7 x, & there are 48 choices for x) So P(flop 4-of-a-kind | pp) = 48/choose(50,3) = 0.245% = 1 in 408.

P(flop an ace high flush)? [where the ace might be on the board] -- 4 suits -- one of the cards must be an ace. choose(12,4) possibilities for the others. So P(flop ace high flush) = 4 * choose(12,4) / choose(52,5) = 0.0762%, or 1 in 1313. P(flop a straight flush)? -- 10 different straight-flushes in each suit. (5 high, 6 high, …, Ace high) So P(flop straight flush) = 4 * 10 / choose(52,5) = 0.00154%, or 1 in 64974.

3) Pot odds and expected value, continued. From previous lecture: to call an all-in, need P(win) > B ÷ (B+pot). Expressed as an odds ratio, this is sometimes referred to as pot odds or express odds. If the bet is not all-in & another betting round is still to come, need P(win) > wager ÷ (wager + winnings), where winnings = pot + amount you’ll win on later betting rounds, wager = total amount you will wager including the current round & later rounds, assuming no folding. The terms Implied-odds / Reverse-implied-odds describe the cases where winnings > pot or where wager > B, respectively. See p66.

4. Yang / Kravchenko. Yang A 10u. Pot is 19million. Bet is 8.55 million. Needs P(win) > 8.55 ÷ (8.55 + 19) = 31%. vs. AA: 8.5%. AJ-AK: 25-27%. KK-TT: 29%. 99-22: 44-48%. KQs: 56%. Bayesian method: average these probabilities, weighting each by its likelihood. See p49-53.