1. Stat 35b: Introduction to Probability with Applications to Poker Outline for the day, Tue 3/13/12: 1.Fred Savage hand. 2.Random walks, continued, for 7.14. 3.Pdfs and cdfs, for 6.12. 4.Optimal play, ch 6.3, pp 109-113. 5.Project B tournament. Homework 4, Stat 35, due Thu March 15, 12:30pm: 6.12, 7.2, 7.8, 7.14. I will discuss the answers in class Thursday. Hw4 probably won’t get graded til Tuesday, so I will just give them back in the final exam. Final Exam: Tuesday, March 20, 11:30 AM - 2:30 PM, here in Royce 156. Thur 3/15 will be review and examples. Be sure to submit your reviews of the course online via my.ucla.edu.

2. 1. Fred Savage hand. 2. Random walks, continued, for 7.14. Theorem 7.6.7, p152, describes another simplified scenario. Suppose you either double each hand you play, or go to zero, each with probability 1/2. Again, P(win a tournament) is prop. to your number of chips. Proof. p 0 = 0, and p 1 = 1/2 p 2 = 1/2 p 2, so again, p 2 = 2 p 1. We have shown that, for j = 0, 1, and 2, p j = j p 1. (induction:) Suppose that, for j ≤ m, p j = j p 1. We will show that p 2m = (2m) p 1. Therefore, p j = j p 1 for all j = 2 k. That is, P(win the tournament) is prop. to number of chips. This time, p m = 1/2 p 0 + 1/2 p 2m. If p j = j p 1 for j ≤ m, then we have mp 1 = 0 + 1/2 p 2m, so p 2m = 2mp 1. Done. 7.14 refers to Theorem 7.6.8, p152. You have k of the n chips in play. Each hand, you gain 1 with prob. p, or lose 1 with prob. q=1-p. Suppose 0<p <1 and p≠0.5. Let r = q/p. Then P(you win the tournament) = (1-r k )/(1-r n ). The proof is again by induction, and is similar to the proof we did last class of Theorem 7.6.6. Notice that if k = 0, then (1-r k )/(1-r n ) = 0. If k = n, then (1-r k )/(1-r n ) = 1.

3. 3. Pdfs and cdfs, for 6.12. Suppose X has the cdf F(c) = P(X ≤ c) = 1 - exp(-4c), for c ≥ 0, and F(c) = 0 for c < 0. Then to find the pdf, f(c), take the derivative of F(c). f(c) = F’(c) = 4exp(-4c), for c ≥ 0, and f(c) = 0 for c<0. Thus, X is exponential with = 4. Now, what is E(X)? E(X) = ∫ -∞ ∞ c f(c) dc = ∫ 0 ∞ c {4exp(-4c)} dc = ¼, after integrating by parts, or just by remembering that if X is exponential with = 4, then E(X) = ¼. For 6.12, the key things to remember are 1. f(c) = F'(c). 2. For an exponential random variable with mean, F(c) = 1 – exp(-1/ c). 3. For any Z, E(Z) = ∫ c f(c) dc. And, V(Z) = E(Z 2 ) – [E(Z)] 2, where E(Z 2 ) = ∫ c 2 f(c) dc. 4. E(X) for exponential = 1/. 5. E(X 2 ) for exponential = 2/ 2. For 7.14, the key things to remember are 1.If p≠0.5, then by Theorem 7.6.8, P(win tournament) = (1-r k )/(1-r n ), where r = q/p. 2.Let x = r 2. If –x 3 + 2x -1 = 0, that means (x-1)(-x 2 - x + 1) = 0. There are 3 solutions to this. One is x = 1. The others occur when x 2 + x - 1 = 0, so x = -1 +/- sqrt(1+4)/2 = -2.118 or 0.118. So, x = -2.118, 0.118, or 1. Two of these possibilities can be ruled out. Remember that p≠0.5.

4. 4. Optimal play, ch 6.3, pp 109-113. 5. Project B tournament.

5. tourn1(name1, decision, myfast1=0)I did 2000 tournaments. Points were 13, 8, 5. The first 3 in class count 20 times normal. 260, 160, 100. The last 1 counts 100 times normal. 1300, 800, 500. Tournaments with errors don’t count. FeelinLucky 6905 + 260 gorgeousgeorge 6596 extracards 5738 AMrfortune 5580 chance 4566 + 100 Jack 4522 + 260 narwhal 4289 ibet 3944 + 160 helios 3838 bomb 2061 dexter 1323 millionaire 1187 + 160 Kidnapped 1074 + 100 lambo 377