Stat 35b: Introduction to Probability with Applications to Poker Outline for the day: 1. Bayes’ Rule again 2.Gold vs. Benyamine 3.Bayes’ Rule example 4.Variance, CLT, and prop bets 5.CLT and pairs Prizes: ten 100 grand bars, pistachio nuts, blowpops, sunflower seeds, choco santas, moonpies,10 butterfingers, 10 crunchbars, marshmallow cookies, Gum (2), trail mix, hershey’s kissables, pez + pencil set, 5 Lakers pens, peanut butter wafers, toffee popcorn, yoyos, spytech markers, set of erasers, Stapler set (mickey, winnie, cars), mint chapstick, play-doh, Clip-on calculator, peppermint candy canes, cards (3 batman, 1 bratz, 2 carebears).   u 

1. Bayes’ Rule If B 1, …, B n are disjoint events with P(B 1 or … or B n ) = 1, then P(B i | A) = P(A | B i ) * P(B i ) ÷ [ ∑P(A | B j )P(B j )]. Ex. Let “disease” mean you really have the disease, and let “+” mean the test says you are positive; “-” means the test says you are negative. Suppose P(disease) = 1%, & the test is 95% accurate: [ P(+ | disease) = 95%, P(- | no disease) = 95% ]. Then what is P(disease | +)? Using Bayes’ rule, P(disease | +) = P(+ | disease) * P(disease) P(+ | disease)P(disease) + P(+ | no disease) P(no disease) = 95% * 1% % * 1% + 5% * 99% = 16.1%.

3. Bayes’ rule example. Suppose P(nuts) = 1%, and P(horrible hand) = 10%. Suppose that P(huge bet | nuts) = 100%, and P(huge bet | horrendous hand) = 30%. What is P(nuts | huge bet)? P(nuts | huge bet) = P(huge bet | nuts) * P(nuts) P(huge bet | nuts) P(nuts) + P(huge bet | horrible hand) P(horrible hand) = 100% * 1% % * 1% + 30% * 10% = 25%.

4. 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 mean µ and a standard deviation 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!

5. CLT and pairs. 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 mean µ and a standard deviation of  ÷√n. Two interesting things about this: (i) As n --> ∞. X --> normal. e.g. average number of pairs per hand, out of n hands. µ = P(pair) = 3/51 = 5.88%. (ii)About 95% of the time, a std normal random variable is within -2 to +2. So 95% of the time, (X - µ) ÷ (  /√n) is within -2 to +2. So 95% of the time, (X - µ) is within -2 (  /√n) to +2 (  /√n). So 95% of the time, X is within µ - 2 (  /√n) to µ + 2 (  /√n). That is, 95% of the time, X is in the interval µ +/- 2 (  /√n).