Stat 35b: Introduction to Probability with Applications to Poker Outline for the day: 1.E(X+Y) = E(X) + E(Y) examples. 2.CLT examples. 3.Lucky poker. 4.Farha vs. Gold 5.Prop bets. 6.Flush draws and straight draws.   u 

1.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 

2) 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.

68% between -1.0 and % between and 1.96

If X 1, X 2, … are iid with expected value µ and sd , then by the CENTRAL LIMIT THEOREM (CLT), (X - µ) ÷ (  /√n) ---> Standard Normal. This term (  /√n) is the SD of X and is sometimes called the Standard Error of X. Why? Var(X) = Var([X 1 + X 2 +…+X n ] ÷ n) = [Var(X 1 )+Var(X 2 )+…+Var(X n )] ÷ n 2 = n  2 ÷ n 2 =  2 ÷ n. So, SD(X) = √ [  2 ÷ n] =  ÷ √n. Suppose a game has mean -25 cents, and SD $5. You play 10,000 times. Assume each game is independent of the others. Let X = your average profit over these 10,000 games. a) What is E(X)? b) What is SD(X)? c) What is P(X > -20 cents)? a)E(X) = µ = -25 cents. b)So SD(X) = $5 ÷ √10,000 = $5 ÷ 100 = 5 cents. c)P(X > -20 cents) = P(X - µ > -20 cents - µ) = P(X - µ > 5 cents) = P{ (X - µ) ÷ (  /√n) > 5 cents ÷ (  /√n)} = P{ (X - µ) ÷ (  /√n) > 1} ~ P(Z > 1) ~ [1-68%] ÷ 2 = 16%.

3. “Lucky Poker” A, B, C. A & B are a team. No strategy: only muck (fold) at the end or don’t muck. First player to two “points” wins the game. If A has a point and B doesn’t, A should show first. B can muck. P(C wins in 2 hands) = 1/9 = 3/27 P(C wins in 3 hands) = 4/27. (ACC, BCC, CAC, CBC) P(C wins in 4 hands) = 1/27. Why? 6 ways for this to happen. {CABC, CBAC, ACBC, BCAC, ABCC, or BACC}. If A has a pt and B doesn’t, then P(B winning a pt on next hand) = 1/6: {ABC, ACB, BAC, BCA, CAB, CBA}. So, P(C wins in 4 hands) = 6 x {1/3 x 1/3 x 1/6 x 1/3} = 1/27. So, P(C wins) = 3/27 + 4/27 + 1/27 = 8/27.

4) High Stakes Poker, Farha vs. Gold

5. Variance, CLT, and prop bets. Central Limit Theorem (CLT): if X 1, X 2 …, X n are iid with mean µ& SD  then (X - µ) ÷ (  /√n) ---> Standard Normal. (mean 0, SD 1). In other words, X has a mean of µ and a SD of  ÷√n. As n increases, (  ÷ √n) decreases. So, the more independent trials, the smaller the SD (and variance) of X. i.e. additional bets decrease the variance of your average. If X and Y are independent, then E(X+Y) = E(X) + E(Y), and V(X+Y) = V(X) + V(Y). Let X = your profit on wager #1, Y = profit on wager #2. If the two wagers are independent, then V(total profit) = V(X) + V(Y) > V(X). So, additional bets increase the variance of your total!