Transcript day9
Stat 35b: Introduction to Probability with Applications to Poker Outline for the day: 1. Collect hw3, return hw2, give out hw4. 2. Negative binomial 3. E(X+Y) 4. Negreanu / Harman, P(flop 2 pairs) 5. Running it twice 6. “Unbeatable” strategy Remember: project A code due Mon 5pm by email! u u 2. Negative Binomial Random Variables. Recall: if each trial is independent, and each time the probability of an occurrence is p, and X = # of trials until the first occurrence, then: X is Geometric (p), P(X = k) = p1 qk - 1, µ = 1/p, s = (√q) ÷ p. Suppose now X = # of trials until the rth occurrence. Then X = negative binomial (r,p). e.g. the number of hands you have to play til you’ve gotten r=3 pocket pairs. Now X could be 3, 4, 5, …, up to ∞. pmf: P(X = k) = choose(k-1, r-1) pr qk - r. e.g. say r=3 & k=7: P(X = 7) = choose(6,2) p3 q4. Why? Out of the first 6 hands, there must be exactly r-1 = 2 pairs. Then pair on 7th. P(exactly 2 pairs on first 6 hands) = choose(6,2) p2 q4. P(pair on 7th) = p. If X is negative binomial (r,p), then µ = r/p, and s = (√rq) ÷ p. e.g. Suppose X = the number of hands til your 12th pocket pair. P(X = 100)? E(X)? s? X = Geometric (12, 5.88%). P(X = 100) = choose(99,11) p12 q88 = choose(99,11) * 0.0588 ^ 12 * 0.9412 ^ 88 = 0.104%. E(X) = r/p = 12/0.0588 = 204.1. s = sqrt(12*0.9412) / 0.0588 = 57.2. So, you’d typically expect it to take 204.1 hands til your 12th pair, +/- around 57.2 hands. 3) E(X+Y) = E(X) + E(Y). Whether X & Y are independent or not! Similarly, E(X + Y + Z + …) = E(X) + E(Y) + E(Z) + … And, if X & Y are independent, then V(X+Y) = V(X) + V(Y). so SD(X+Y) = √[SD(X)^2 + SD(Y)^2]. Example 1: Play 10 hands. X = your total number of pocket aces. What is E(X)? X is binomial (n,p) where n=10 and p = 0.00452, so E(X) = np = 0.0452. Alternatively, X = # of pocket aces on hand 1 + # of pocket aces on hand 2 + … So, E(X) = Expected # of AA on hand1 + Expected # of AA on hand2 + … Each term on the right = 1 * 0.00452 + 0 * 0.99548 = 0.00452. So E(X) = 0.00452 + 0.00452 + … + 0.00452 = 0.0452. Example 2: Play 1 hand, against 9 opponents. X = total # of pocket aces at the table. Now what is E(X) Note: not independent! If you have AA, then it’s unlikely anyone else does too. Nevertheless, Let X1 = 1 if player #1 has AA, and 0 otherwise. X2 = 1 if player #2 has AA, and 0 otherwise. etc. Then X = X1 + X2 + … + X10. So E(X) = E(X1) + E(X2) + … + E(X10) = 0.00452 + 0.00452 + … + 0.00452 = 0.0452. 4. Harman/Negreanu 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. 5) Harman / Negreanu, and running it twice. Harman has 10 7 . Negreanu has K Q . The flop is 10u 7 Ku . Harman’s all-in. $156,100 pot. P(Negreanu wins) = 29%. P(Harman wins) = 71%. Let X = amount Harman has after the hand. If they run it once, E(X) = $0 x 29% + $156,100 x 71% = $110,831. If they run it twice, what is E(X)? There’s some probability p1 that Harman wins both times ==> X = $156,100. There’s some probability p2 that they each win one ==> X = $78,050. There’s some probability p3 that Negreanu wins both ==> X = $0. E(X) = $156,100 x p1 + $78,050 x p2 + $0 x p3. If the different runs were independent, then p1 = P(Harman wins 1st run & 2nd run) would = P(Harman wins 1st run) x P(Harman wins 2nd run) = 71% x 71% = 50.4%. But, they’re not quite independent! Very hard to compute p1 and p2. However, you don’t need p1 and p2 ! X = the amount Harman gets from the 1st run + amount she gets from 2nd run, so E(X) = E(amount Harman gets from 1st run) + E(amount she gets from 2nd run) = $78,050 x P(Harman wins 1st run) + $0 x P(Harman loses first run) + $78,050 x P(Harman wins 2nd run) + $0 x P(Harman loses 2nd run) = $78,050 x 71% + $0 x 29% + $78,050 x 71% + $0 x 29% = $110,831. HAND RECAP: Harman 10 7 . Negreanu K Q . Flop 10u 7 Ku . Harman’s all-in. $156,100 pot. P(Negreanu wins) = 29%. P(Harman wins) = 71%. --------------------------------------------------------------------------The standard deviation changes a lot! Say they run it once. V(X) = E(X2) - µ2. µ = $110,831, so µ2 ~ $12.3 billion. E(X2) = ($156,1002)(71%) + (02)(29%) = $17.3 billion. So V(X) = $17.3 billion - $12.3 billion = $5 billion. The standard deviation s = sqrt($5 billion) ~ $71,000. So if they run it once, Harman expects to get back about $110,831 +/- $71,000. If they run it twice? Hard to compute, but approximately, if each run were independent, then V(X1+X2) = V(X1) + V(X2), so if X1 = amount she gets back on 1st run, and X2 = amount she gets from 2nd run, then V(X1+X2) ~ V(X1) + V(X2) ~ $1.3 billion + $1.3 billion = $2.6 billion, The standard deviation s = sqrt($2.6 billion) ~ $51,000. So if they run it twice, Harman expects to get back about $110,831 +/- $51,000. 6) “Unbeatable Texas Holdem Strategy” http://www.freepokerstrategy.com : all in with AK-AT or any pair. P(getting such a hand) = 4 x [16/1326] + 13 x [6/1326] = 4 x 1.2% + 13 x 0.45% = 10.7%. Say you’re dealt 100 hands. Pay ~11 blinds = $55. Expect 10.7 (~ 11) such good hands. Say you’re called by 88-AA, and AK, for $100 on avg. P(player 1 has one of these) = 7 x 0.45% + 1.2% = 4.4%. P(of 8 opponents, someone has one of these) ~ 1 - (95.6%)8 = 30%. So, you win pre-flop 70% of the time. (Say $10 on avg.) = 11 x 70% x $10 = $77 profit. Other 30%, you’re on avg about a 65-35 underdog, so you win 11 x 30% x 35% x $100 = $115.50 lose 11 x 30% x 65% x $100 = $214.50. Total: exp. to win $77 + $115.50 - $55 - $214.50 = -$77/ 100 hands.