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Stat 35b: Introduction to Probability with Applications to Poker Outline for the day:

Stat 35b: Introduction to Probability with Applications to Poker Outline for the day: P(flop two pairs). E(X+Y) = E(X) + E(Y) example. Farha vs. Gold. Bayes’s rule. Teams for project B. Continuous random variables. Uniform, normal, and standard normal distributions.

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Stat 35b: Introduction to Probability with Applications to Poker Outline for the day:

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  1. Stat 35b: Introduction to Probability with Applications to Poker Outline for the day: • P(flop two pairs). • E(X+Y) = E(X) + E(Y) example. • Farha vs. Gold. • Bayes’s rule. • Teams for project B. • Continuous random variables. • Uniform, normal, and standard normal distributions.  u   u 

  2. 1. P(flop two pairs). If you’re sure to be all-in next hand,what is P(you will flop two pairs)? Tough one. Don’t double-count (4 4 9 9 Q) and (9 9 4 4 Q)! There are choose(13,2) possibilities for the NUMBERS of the two pairs. For each such choice (such as 4 & 9), there are choose(4,2) choices for the suits of the lower pair, and same for the suits of the higher pair. So, choose(13,2) * choose(4,2) * choose(4,2) different possibilities for the two pairs. For each such choice, there are 44 [52 - 8 = 44] different possibilities for your fifth card, so that it’s not a full house but simply two pairs. So, P(flop two pairs) = choose(13,2) * choose(4,2) * choose(4,2) * 44 / choose(52,5) ~ 4.75%, or 1 in 21.

  3. 2.E(X+Y) = E(X) + E(Y) example.Deal the cards face up, without reshuffling. Let Z = the number of cards til the 2nd king. What is E(Z)?  u   u 

  4. Z = the number of cards til the 2nd king. What is E(Z)? Let X1 = number of cards til 1st king. Let X2 = number of cards after 1st king til 2nd king. Let X3 = number of cards after 2nd king til 3rd king. Let X4 = number of cards after 3rd king til 4th king. Let X5 = number of cards after 4th king til the end of the deck. Clearly, X1 + X2 + X3 + X4 + X5 = 52. By symmetry, E(X1) = E(X2) = E(X3) = E(X4) = E(X5). Therefore, E(X1) = E(X2) = 52/5. Z = X1 + X2, so E(Z) = E(X1) + E(X2) = 2 x 52/5 = 20.8.

  5. 3. Gold vs. Farha. 4. Bayes’s rule, p49-52. Suppose that B1, B2 , Bn are disjoint events and that exactly one of them must occur. Suppose you want P(B1 | A), but you only know P(A | B1 ), P(A | B2 ), etc., and you also know P(B1), P(B2), …, P(Bn). Bayes’ Rule: If B1 , …, Bn are disjoint events with P(B1 or … or Bn) = 1, then P(Bi | A) = P(A | Bi ) * P(Bi) ÷ [ ∑P(A | Bj)P(Bj)]. Why? Recall: P(X | Y) = P(X & Y) ÷ P(Y). So P(X & Y) = P(X | Y) * P(Y). P(B1 | A) = P(A & B1 ) ÷ P(A) = P(A & B1 ) ÷ [ P(A & B1) + P(A & B2) + … + P(A & Bn) ] = P(A | B1 ) * P(B1) ÷ [ P(A | B1)P(B1) + P(A | B2)P(B2) + … + P(A & Bn)P(Bn) ].

  6. Bayes’s rule, continued. Bayes’s rule: If B1 , …, Bn are disjoint events with P(B1 or … or Bn) = 1, then P(Bi | A) = P(A | Bi ) * P(Bi) ÷ [ ∑P(A | Bj)P(Bj)]. See example 3.4.1, p50. If a test is 95% accurate and 1% of the pop. has a condition, then given a random person from the population, P(she has the condition | she tests positive) = P(cond | +) = P(+ | cond) P(cond) ÷ [P(+ | cond) P(cond) + P(+ | no cond) P(no cond)] = 95% x 1% ÷ [95% x 1% + 5% x 99%] ~ 16.1%. Tests for rare conditions must be extremely accurate.

  7. Back to Gold vs. Farha. Gold: 10u 7 Farha: Qu Q Flop: 9u 8 7 Who really is the favorite (ignoring all other players’ cards)? Gold’s outs: J, 6, 10, 7. (4 + 4 + 3 + 2 = 13 outs, 32 non-outs) P(Gold wins) = P(Out Out or Jx [x ≠ 10] or 6x or 10y [y≠Q,9,8] or 7z [z≠Q]) = [choose(13,2) + 4*28 + 4*32 + 3*24 + 2*30] ÷ choose(45,2) = 450 ÷ 990 = 45.45%. What would you guess Gold had? Say he’d do that 50% of the time with a draw, 100% of the time with an overpair, and 90% of the time with two pairs. (and that’s it) Using Bayes’ rule, P(Gold has a DRAW | Gold raises ALL-IN) = . [P(all-in | draw) * P(draw)] . [P(all-in | draw) P(draw)] + [P(all-in | overpair) P(overpair)] + [P(all-in | 2pairs) P(2 pairs)] = [50% * P(draw)] ÷ [50% * P(draw)] + [100% * P(overpair)] + [90% * P(2 pairs)]

  8. 5. TEAMS FOR PROJECT B. See teams.txt. Project B must be emailed to me by Friday, March 9, 8pm. team a Jun Wai team b Zesheng Benham team c Yoo Kwangchae team d Sungjae Sukhmeet team e Delroy Woosung team f James Logan team g Sarkis Deaeun team h Kelly Boris team i Michelle Te team j Yuk Noel team k Steven Nicholas team l Chi Hoi team m Bonnie Pin-Han team n Cody Dahhee

  9. 6. Continuous random variables and their densities, p103-107. Density (or pdf = Probability Density Function) f(y): ∫B f(y) dy = P(X in B). Expected value, µ = E(X) = ∫ y f(y) dy. (= ∑ y P(y) for discrete X.) Variance, s2 = V(X) = E(X2) – m2. SD(X) = √V(X). For examples of pdfs, see p104, 106, and 107.

  10. 7. Examples: uniform, normal, and standard normal variables. * Uniform (0,1). See p107-109. f(y) = 1, for y in (0,1). µ = 0.5. s ~ 0.29. P(X is between 0.4 and 0.6) = ∫.4 .6 f(y) dy = ∫.4 .6 1 dy = 0.2. * Exponential (l). See p114. f(y) = le-ly, for y ≥ 0. E(X) = 1/l. SD(X) = 1/l. * Normal. pp 115-117. mean = µ, SD = s, f(y) = 1/√(2πs2) e-(y-µ)/2s2. Symmetric around µ, 50% of the values are within 0.674 SDs of µ, 68.27% of the values are within 1 SD of µ, and 95% are within 1.96 SDs of µ. * Standard Normal. Normal with µ = 0, s = 1. See pp 117-118.

  11. Standard normal density: 68.27% between -1.0 and 1.0 95% between -1.96 and 1.96

  12. Project B example. zelda = function(numattable1, crds1, board1, round1, currentbet, mychips1, pot1, roundbets, blinds1, chips1, ind1, dealer1, tablesleft){ a1 = 0 ## how much I'm gonna end up betting. Note that the default is zero. a2 = min(mychips1, currentbet) ## how much it costs to call if(round1 == 1){ ## pre-flop: ## AK: Make a big raise if nobody has yet. Otherwise call. ## AQ: call a small raise, or make one if nobody has yet. ## AJ, AT, KQ, KJ, QJ: call a tiny raise. ## A9, KT, K9, QT, JT, T9: call a tiny raise if in late position (within 2 of the dealer). ## Suited A2-AJ: call a small raise. ## 22-99: call a small raise. ## TT-KK: make a huge raise. If someone's raised huge already, then go all in. ## AA: make a small raise. If there's been a raise already, then double how much it is to you. a3 = 2*blinds1+1 ## how much a tiny raise would be a4 = 4*blinds1+1 ## how much a small raise would be a5 = max(8*blinds1,mychips1/4)+1 ## how much a big raise would be a6 = max(12*blinds1,mychips1/2)+1 ## how much a huge raise would be a7 = dealer1 - ind1 if(a7 < -.5) a7 = a7 + numattable1 ## your position: a7 = how many hands til you're dealer if((crds1[1,1] == 14) && (crds1[2,1] == 13)){a1 = max(a2,a5) }

  13. if((crds1[1,1] == 14) && (crds1[2,1] == 12)){ if(a2 < a4){ a1 = a4 } else if(a2 > a5){ a1 = 0 } else a1 = a2 } if(((crds1[1,1] == 14) && ((crds1[2,1] < 11.5) && (crds1[2,1] > 9.5))) || ((crds1[1,1] == 13) && (crds1[2,1] > 10.5)) || ((crds1[1,1] == 12) && (crds1[2,1] == 11))){ if(a2 < a3) a1 = a2 } if(((crds1[1,1] == 14) && (crds1[2,1] == 9)) || ((crds1[1,1] == 13) && ((crds1[2,1] == 10) || (crds1[2,1] == 9))) || ((crds1[1,1] == 12) && (crds1[2,1] == 10)) || ((crds1[1,1] == 11) && (crds1[2,1] == 10)) || ((crds1[1,1] == 10) && (crds1[2,2] == 9))){ if((a2 < a3) && (a7<2.5)) a1 = a2 } if((crds1[1,2] == crds1[2,2]) && (crds1[1,1] == 14) && (crds1[2,1] < 11.5)){ if(a2<a4) a1 = a2 ## Note: this trumps the previous section, since it comes later in the code. }

  14. if((crds1[1,1] == crds1[2,1])){ ## pairs: if(crds1[1,1] < 9.5){ if(a2 < a4) a1 = a2 } else if(crds1[1,1] < 13.5){ if(a2<a5) a1 = a5 else a1 = mychips1 } else { if(a2 < blinds1 + .5) a1 = a4 else a1 = min(2*a2,mychips1) }} } if(round1 == 2){ ## post-flop: ## If there's a pair on the board and you don't have a set, then check/call up to small bet. ## Same thing if there's 3-of-a-kind on the board and you don't have a full house or more. ## If you have top pair or an overpair or two pairs or a set, make a big bet (call any bigger bet). ## Otherwise, if nobody's made even a small bet yet, then with prob. 20% make a big bluff bet. ## If you're the last to decide and nobody's bet yet, then increase this prob. to 50%. ## If you have an inside straight draw or flush draw then make a small bet (call any bigger bet). ## If you have a straight or better, then just call. ## Otherwise fold. a5 = min(sum(roundbets[,1]),mychips1) ## how much big bet would be (prev round's pot size) a6 = min(.5*sum(roundbets[,1]),mychips1) ## how much a small bet would be x = handeval(c(crds1[1:2,1], board1[1:3,1]), c(crds1[1:2,2], board1[1:3,2])) ## what you have x1 = handeval(c(board1[1:3,1]),c(board1[1:3,2])) ## what's on the board y = straightdraw1(c(crds1[1:2,1], board1[1:3,1]))

  15. z = flushdraw1(c(crds1[1:2,2], board1[1:3,2])) topcard1 = max(board1[1:3,1]) a7 = runif(1) ## random number uniformly distributed between 0 and 1 a8 = (1:numattable1)[roundbets[,1] == roundbets[ind1,1]] ## others who can still bet with you ## The next 5 lines may seem weird, but the purpose is explained in the next comment: a9 = a8 - dealer1 for(i in 1:length(a9)) if(a9[i]<.5) a9[i] = a9[i] + numattable1 a10 = ind1 - dealer1 if(a10 < .5) a10 = a10 + numattable1 a11 = 2*(a10 == max(a9)) ## So a11 = 2 if you're last to decide; otherwise a11 = 0. if((x1 > 1000000) && (x < 3000000)){ if(a2 < a6) a1 = a2 } else if((x1 > 3000000) && (x < 6000000)){ if(a2 < a6) a1 = a2 } else if(x > 1000000 + 15^3*topcard1){ a1 = max(a5,a2) } else if((a2 < a6) && ((a7 < .20) || ((a7 < .50) && (a11>1)))){ a1 = a6 } if((y == 4) || (z == 4)) a1 = max(a6, a2) if(x > 4000000) a1 = a2 }

  16. if(round1 == 3){ ## after turn: ## If there's a pair on the board and you don't have a set, then check/call up to small bet. ## Same thing if there's 3-of-a-kind on the board and you don't have a full house or more. ## Otherwise, if you have top pair or better, go all in. ## If you had top pair or overpair but now don't, then check/call a medium bet but fold to more. ## If you have an inside straight draw or flush draw then check/call a medium bet as well. ## Otherwise check/fold. a6 = min(1/3*sum(roundbets[,1:2]),mychips1) ## small bet (1/3 of prev round's pot size) a5 = min(.75*sum(roundbets[,1:2]),mychips1) ## medium bet (3/4 of prev round's pot size) x = handeval(c(crds1[1:2,1], board1[1:4,1]), c(crds1[1:2,2], board1[1:4,2])) ## what you have x1 = handeval(c(board1[1:4,1]),c(board1[1:4,2])) ## what's on the board y = straightdraw1(c(crds1[1:2,1], board1[1:4,1])) z = flushdraw1(c(crds1[1:2,2], board1[1:4,2])) topcard1 = max(board1[1:4,1]) oldtopcard1 = max(board1[1:3,1]) if((x1 > 1000000) && (x < 3000000)){ if(a2 < a6) a1 = a2 } else if((x1 > 3000000) && (x < 6000000)){if(a2 < a6) a1 = a2 } else if(x > 1000000 + 15^3*topcard1){ a1 = mychips1 } else if(x > 1000000 + 15^3*oldtopcard1){ if(a2 < a5) a1 = a2 } else if((y == 4) || (z == 4)){ if(a2 < a5) a1 = a2 } }

  17. if(round1 == 4){ ## after river: ## If there's a pair on the board and you don't have a set, then check/call up to small bet. ## Same thing if there's 3-of-a-kind on the board and you don't have a full house or more. ## Otherwise, if you have two pairs or better, go all in. ## If you have one pair, then check/call a small bet. ## With nothing, go all-in with probability 10%; otherwise check/fold. a6 = .45+runif(1)/10 ## random number between .45 and .55 a5 = min(a6*sum(roundbets[,1:3]),mychips1) ## small bet~ 1/2 of pot size; varies randomly x = handeval(c(crds1[1:2,1], board1[1:5,1]), c(crds1[1:2,2], board1[1:5,2])) x1 = handeval(c(board1[1:5,1]),c(board1[1:5,2])) ## what's on the board if((x1 > 1000000) && (x < 3000000)){ if(a2 < a5) a1 = a2 } else if((x1 > 3000000) && (x < 6000000)){ if(a2 < a5) a1 = a2 } else if(x > 2000000){ a1 = mychips1 } else if(x > 1000000){ if(a2 < a5) a1 = a2 } else if(runif(1)<.10){ a1 = mychips1 } } round(a1) } ## end of zelda

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