Stat 35b: Introduction to Probability with Applications to Poker Outline for the day: 1.Review List 2.Review of Discrete variables 3.Nguyen / Szenkuti 4.Hansen / Martens 5.Sums of random variables 6.Farha/Antonius 7.Continuous Random Variables, Density, Uniform, Normal 8.LLN & CLT 9.Hansen / Martens For the midterm Monday: Bring a calculator! All notes are ok.   u 

Review List: Axioms of probability. Variance and SD. Multiplication rule of counting. Uniform Random Variables. Permutations and Combinations. Bernoulli RVs. Addition Rule of probability. Binomial RVs. Conditional probability and Independence. Geometric RVs. Multiplication rule of probability. Negative binomial RVs. Counting problems and tricks. E(X+Y). Odds ratios. Random variables, pmf. Expected value. Pot odds calculations.

Discrete Variables: Bernoulli. 0/1. f(1) = p, f(0) = q. E(X) = p.  = √(pq). Binomial.# of successes out of n independent tries. f(k) = choose(n, k) * p k q n-k. E(X) = np.  = √(npq). Geometric.# of (independent) tries until the first success. f(k) = p 1 q k-1. E(X) = 1/p.  = (√q) ÷ p. Neg. Binomial.# of (independent) tries until the rth success. f(k) = choose(k-1, r-1) p r q k-r. E(X) = r/p.  = (√rq) ÷ p.

11/4/05, Travel Channel, World Poker Tour, $1 million Bay 101 Shooting Star. 4 players left, blinds $20,000 / $40,000, with $5,000 antes. Avg stack = $1.1 mil. 1st to act: Danny Nguyen, A  7 . All in for $545,000. Next to act: Shandor Szentkuti, A  K . Call. Others (Gus Hansen & Jay Martens) fold.(66% - 29%). Flop: 5 K 5 .(tv 99.5%; cardplayer.com: 99.4% - 0.6%). P(tie) = P(55 or A5) = (1 + 2*2) ÷ choose(45,2) = 0.505%. 1 in 198. P(Nguyen wins) = P(77) = choose(3,2) ÷ choose(45,2) = 0.30%. 1 in 330. [Note: tv said “odds of running 7’s on the turn and river are 274:1.” Given Hansen/Martens’ cards, choose(3,2) ÷ choose(41,2) = 1 in ] * Szentkuti was eliminated next hand, in 4th place. Nguyen went on to win it all. Turn: 7 . River: 7  !

11/4/05, Travel Channel, World Poker Tour, $1 million Bay 101 Shooting Star. 3 players left, blinds $20,000 / $40,000, with $5,000 antes. Avg stack = $1.4 mil. (pot = $75,000) 1st to act: Gus Hansen, K  9 . Raises to $110,000. (pot = $185,000) Small blind: Dr. Jay Martens, A  Q. Re-raises to $310,000. (pot = $475,000) Big blind: Danny Nguyen, 7  3 . Folds. Hansen calls. (tv: 63%-36%.) (pot = $675,000) Flop: 4  9 6 .(tv: 77%-23%; cardplayer.com: 77.9%-22.1%) P(no A nor Q on next 2 cards) = 37/43 x 36/42 = 73.8% P(AK or A9 or QK or Q9) = ( ) ÷ (43 choose 2) = 3.3% So P(Hansen wins) = 73.8% + 3.3% = 77.1%. P(Martens wins) = 22.9%.

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]. Also, if Y = 9X, then E(Y) = 9E(Y), and SD(Y) = 9SD(X). V(Y) = 81V(X). Farha vs. Antonius…. Running it 4 times. Let X = chips you have after the hand. Let p be the prob. you win. X = X 1 + X 2 + X 3 + X 4, where X 1 = chips won from the first “run”, etc. E(X) = E(X 1 ) + E(X 2 ) + E(X 3 ) + E(X 4 ) = 1/4 pot (p) + 1/4 pot (p) + 1/4 pot (p) + 1/4 pot (p) = pot (p) = same as E(Y), where Y = chips you have after the hand if you ran it once!!! But the SD is smaller: clearly X 1 = Y/4, so SD(X 1 ) = SD(Y)/4. So, V(X 1 ) = V(Y)/16. V(X) ~ V(X 1 ) + V(X 2 ) + V(X 3 ) + V(X 4 ), = 4 V(X 1 ) = 4 V(Y) / 16 = V(Y) / 4. So SD(X) = SD(Y) / 2.

Continuous Random Variables, Density, Uniform, Normal Density (or pdf = Probability Density Function) f(y): ∫ B f(y) dy = P(X in B). Expected value (µ) = ∫ y f(y) dy. (= ∑ y P(y) for discrete X.) Example 1: Uniform (0,1). f(y) = 1, for y in (0,1). µ = 0.5.  = P(X is between 0.4 and 0.6) = ∫.4.6 f(y) dy = ∫ dy = 0.2. Example 2: Normal. mean = µ, SD = , 68% of the values are within 1 SD of µ 95% are within 2 SDs of µ Example 3: Standard Normal. Normal with µ = 0,  = 1.

95% between and 1.96

Law of Large Numbers, CLT Sample mean (X) = ∑X i / n iid: independent and identically distributed. Suppose X 1, X 2, etc. are iid with expected value µ and sd , LAW OF LARGE NUMBERS (LLN): X ---> µ. CENTRAL LIMIT THEOREM (CLT): (X - µ) ÷ (  /√n) ---> Standard Normal. Useful for tracking results. Note: LLN does not mean that short-term luck will change. Rather, that short-term results will eventually become negligible.

95% between and 1.96

Truth: -49 or 51, each with prob. 1/2. exp. value = 1.0

Truth: -49 to 51, exp. value = 1.0 Estimated as X +/  /√n =.95 +/- 0.28

* Poker has high standard deviation. Important to keep track of results. * Don’t just track ∑X i. Track X +/  /√n. Make sure it’s converging to something positive.

1st to act: Gus Hansen, K  9 . Raises to $110,000. (pot = $185,000) Small blind: Dr. Jay Martens, A  Q. Re-raises to $310,000. (pot = $475,000) Hansen calls. (pot = $675,000) Flop: 4  9 6 . P(Hansen wins) = 77.1%. P(Martens wins) = 22.9%. Martens checks. Hansen all-in for $800,000 more. (pot = $1,475,000) Martens calls. (pot = $2,275,000) Vince Van Patten: “The doctor making the wrong move at this point. He still can get lucky of course.” Was it the wrong move? His prob. of winning should be ≥ $800,000 ÷ $2,275,000 = 35.2%. Here it was 22.9%. So, if Martens knew what cards Hansen had, he’d be making the wrong move. But given all the possibilities, should he assume he had a 35.2% chance to win? [Harrington: P(bluff) is always ≥ 10%.] * Turn: A  ! River: 2 . * Hansen was eliminated 2 hands later, in 3rd place. Martens then lost to Nguyen.