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Learn about probability with applications to poker, including examples of expected values, the Central Limit Theorem, and practical tips for tracking results and making informed decisions during the game.
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Stat 35b: Introduction to Probability with Applications to Poker Outline for the day: E(X+Y) = E(X) + E(Y) examples. CLT examples. Lucky poker. Farha vs. Gold Prop bets. Flush draws and straight draws. u u
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
2) Law of Large Numbers, CLT Sample mean (X) = ∑Xi / n iid: independent and identically distributed. Suppose X1, X2 , etc. are iid with expected value µ and sd s , LAW OF LARGE NUMBERS (LLN): X ---> µ . CENTRAL LIMIT THEOREM (CLT): (X - µ) ÷ (s/√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 1.0 95% between -1.96 and 1.96
If X1, X2 , … are iid with expected value µ and sd s, then by the • CENTRAL LIMIT THEOREM (CLT), (X - µ) ÷ (s/√n) ---> Standard Normal. • This term (s/√n) is the SD of X and is sometimes called the Standard Error of X. • Why? Var(X) = Var([X1+ X2 +…+Xn ] ÷ n) = [Var(X1)+Var(X2)+…+Var(Xn)] ÷ n2 • = n s2 ÷ n2 = s2 ÷ n. So, SD(X) = √ [s2 ÷ n] = s ÷ √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)? • E(X) = µ = -25 cents. • 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) ÷ (s/√n) > 5 cents ÷ (s/√n)} = P{ (X - µ) ÷ (s/√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.
5. Variance, CLT, and prop bets. Central Limit Theorem (CLT): if X1 , X2 …, Xn are iid with mean µ& SD s, then (X - µ) ÷ (s/√n) ---> Standard Normal. (mean 0, SD 1). In other words, X has a mean of µ and a SD ofs÷√n. As n increases, (s ÷ √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!
