2. 1)Hand in HW. 2)No class Tuesday (Veteran’s Day) 3)Midterm Thursday (1 page, double-sided, of notes allowed) 4)Review List 5)Review of Discrete variables 6)Play (and lose) like the Pros 7)Savage / Tyler 8)P(full house), P(rainbow flop) 9)Nguyen / Szenkuti 10)Hansen / Martens   u 

3. Review List: Axioms of probability. Variance and SD. Multiplication rule of counting. Uniform Random Variables. Permutations and Combinations. Bernoulli RVs. Addition Rule of probability. Binomial RVs. Conditional probability and Independence. Geometric RVs. Multiplication rule of probability. Negative binomial RVs. Counting problems and tricks. E(X+Y). Odds ratios. Random variables, pmf. Expected value. Pot odds calculations.

4. Discrete Variables: Bernoulli. 0/1. f(1) = p, f(0) = q. E(X) = p.  = √(pq). Binomial.# of successes out of n independent tries. f(k) = choose(n, k) * p k q n-k. E(X) = np.  = √(npq). Geometric.# of (independent) tries until the first success. f(k) = p 1 q k-1. E(X) = 1/p.  = (√q) ÷ p. Neg. Binomial.# of (independent) tries until the rth success. f(k) = choose(k-1, r-1) p r q k-r. E(X) = r/p.  = (√rq) ÷ p.

5. Phil Helmuth, Play Poker Like the Pros, Collins, 2003. Strategy for beginners: AA, KK, QQ, or AK. P(getting one of these hands)? 3 x choose(4,2)/choose(52,2) + 4x4/choose(52,2) = 3 x 6/1326 + 16/1326 = 3 x 0.45% + 1.21% = 2.56% = 1 in 39. Say you play $100 NL, table of 9, blinds 2/3, for 39 x 9 = 351 hands. Pay 5 x 39 = 195 dollars in blinds. Expect to play 9 hands. Say P(win preflop) ~ 50%, and in those hands you win ~ $8. Other 50%, always vs. 1 opponent, 60% to win $100. So, expected winnings after 351 hands = -$195 + 9 x 50% x $8 + 9 x 50% x 60% x $100 + 9 x 50% x 40% x -$100 = -$69. That is, you lose $69 every 351 hands on average = $20 per 100 hands.

6. Savage / Tyler. P(flop a full house)? (If you’re all in next hand, no matter what cards you get. Key idea: forget order! Consider all combinations of your 2 cards and the flop. P(flop full house) = # of different full houses / choose(52,5) 13 * choose(4,3) different choices for the triple. For each such choice, there are 12 * choose(4,2) choices left for the pair. So, P(flop full house) = 13 * choose(4,3) * 12 * choose(4,2) / choose(52,5) ~ 0.144%, or 1 in 694.

7. Rainbow board. P(Rainbow flop) = choose(4,3) * 13 * 13 * 13 ÷ choose(52,3) choices for the 3 suits numbers on the 3 cards possible flops ~ 39.76%. Alternative way: conceptually, order the flop cards. No matter what flop card #1 is, P(suit of #2 ≠ suit of #1 & suit of #3 ≠ suits of #1 and #2) = P(suit #2 ≠ suit #1) * P(suit #3 ≠ suits #1 and #2 | suit #2 ≠ suit #1) = 39/51 * 26/50 ~ 39.76%.

8. 11/4/05, Travel Channel, World Poker Tour, $1 million Bay 101 Shooting Star. 4 players left, blinds $20,000 / $40,000, with $5,000 antes. Avg stack = $1.1 mil. 1st to act: Danny Nguyen, A  7 . All in for $545,000. Next to act: Shandor Szentkuti, A  K . Call. Others (Gus Hansen & Jay Martens) fold.(66% - 29%). Flop: 5 K 5 .(tv 99.5%; cardplayer.com: 99.4% - 0.6%). P(tie) = P(55 or A5) = (1 + 2*2) ÷ choose(45,2) = 0.505%. 1 in 198. P(Nguyen wins) = P(77) = choose(3,2) ÷ choose(45,2) = 0.30%. 1 in 330. [Note: tv said “odds of running 7’s on the turn and river are 274:1.” Given Hansen/Martens’ cards, choose(3,2) ÷ choose(41,2) = 1 in 273.3.] * Szentkuti was eliminated next hand, in 4th place. Nguyen went on to win it all. Turn: 7 . River: 7  !

9. 11/4/05, Travel Channel, World Poker Tour, $1 million Bay 101 Shooting Star. 3 players left, blinds $20,000 / $40,000, with $5,000 antes. Avg stack = $1.4 mil. (pot = $75,000) 1st to act: Gus Hansen, K  9 . Raises to $110,000. (pot = $185,000) Small blind: Dr. Jay Martens, A  Q. Re-raises to $310,000. (pot = $475,000) Big blind: Danny Nguyen, 7  3 . Folds. Hansen calls. (tv: 63%-36%.) (pot = $675,000) Flop: 4  9 6 .(tv: 77%-23%; cardplayer.com: 77.9%-22.1%) P(no A nor Q on next 2 cards) = 37/43 x 36/42 = 73.8% P(AK or A9 or QK or Q9) = (9+6+9+6) ÷ (43 choose 2) = 3.3% So P(Hansen wins) = 73.8% + 3.3% = 77.1%. P(Martens wins) = 22.9%.

10. 1st to act: Gus Hansen, K  9 . Raises to $110,000. (pot = $185,000) Small blind: Dr. Jay Martens, A  Q. Re-raises to $310,000. (pot = $475,000) Hansen calls. (pot = $675,000) Flop: 4  9 6 . P(Hansen wins) = 77.1%. P(Martens wins) = 22.9%. Martens checks. Hansen all-in for $800,000 more. (pot = $1,475,000) Martens calls. (pot = $2,275,000) Vince Van Patten: “The doctor making the wrong move at this point. He still can get lucky of course.” Was it the wrong move? His prob. of winning should be ≥ $800,000 ÷ $2,275,000 = 35.2%. Here it was 22.9%. So, if Martens knew what cards Hansen had, he’d be making the wrong move. But given all the possibilities, should he assume he had a 35.2% chance to win? [Harrington: P(bluff) is always ≥ 10%.] * Turn: A  ! River: 2 . * Hansen was eliminated 2 hands later, in 3rd place. Martens then lost to Nguyen.