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Introduction to Probability with Applications to Poker

This outline covers the basics of probability, including sums of random variables, continuous random variables, and the Law of Large Numbers and Central Limit Theorem. It includes examples and applications to poker.

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Introduction to Probability with Applications to Poker

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  1. Stat 35b: Introduction to Probability with Applications to Poker Outline for the day: Sums of random variables Farha/Antonius Continuous Random Variables, Density, Uniform, Normal LLN & CLT.  u   u 

  2. 1) 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). 2) Farha vs. Antonius. Running it 4 times. Let X = chips you have after the hand. Let p be the prob. you win. X = X1 + X2 + X3 + X4, where X1 = chips won from the first “run”, etc. E(X) = E(X1) + E(X2) + E(X3) + E(X4) = 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 X1 = Y/4, so SD(X1) = SD(Y)/4. So, V(X1) = V(Y)/16. V(X) ~ V(X1) + V(X2) + V(X3) + V(X4), = 4 V(X1) = 4 V(Y) / 16 = V(Y) / 4. So SD(X) = SD(Y) / 2.

  3. 3) 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. s = 0.29. P(X is between 0.4 and 0.6) = ∫.4 .6 f(y) dy = ∫.4 .6 1 dy = 0.2. Example 2: Normal. mean = µ, SD = s, 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, s = 1.

  4. 95% between -1.96 and 1.96

  5. 4) 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.

  6. 95% between -1.96 and 1.96

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

  8. Truth: -49 to 51, exp. value = 1.0 Estimated as X +/- 1.96 s/√n = .95 +/- 0.28

  9. * Poker has high standard deviation. Important to keep track of results. * Don’t just track ∑Xi. Track X +/- 1.96 s/√n . Make sure it’s converging to something positive.

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