Stat 35b: Introduction to Probability with Applications to Poker Outline for the day: Collect hw3, give out hw4, no class Thur Oct 29! Play (and lose) like the pros Savage / Tyler P(flop a full house) P(Rainbow flop) Szenkuti / Nguyen Remember: project A code due Mon 8pm by email!   u    u 

2. Play (and lose) like the pros. Phil Hellmuth, 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 39x9 = 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.

3. Savage / Tyler. 4. 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. Notice the difference between this calculation and the probability of flopping 2 pairs. Here, it’s 13 * 12 possibilities for the number on the triplet and the pair instead of choose(13,2) possibilities for the numbers on the two pairs.

5. Rainbow board = all different suits. P(rainbow flop)? How many ways can this happen? Pick 3 suits. For each, 13 possible cards. 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%.

6. Nguyen / Szenkuti. 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.] TURN: 7. River 7! Szentkuti was eliminated next hand, in 4th place. Nguyen went on to win it all.

7. Martens / Hansen. 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%.

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, wrong move if he knew what Hansen had. Turn: A! River: 2. Hansen was eliminated 2 hands later, in 3rd place. Martens then lost to Nguyen.