2. Stat 35b: Introduction to Probability with Applications to Poker Poker Code competition: all-in or fold.   u 

3. 5) P(SOMEONE has AA, given you have KK)? Out of your 8 opponents? Note that given that you have KK, P(player 2 has AA & player 3 has AA) = P(player 2 has AA) x P(player 3 has AA | player 2 has AA) = choose(4,2) / choose(50,2) x 1/choose(48,2) = 0.0000043, or 1 in 230,000. So, very little overlap! Given you have KK, P(someone has AA) = P(player2 has AA or player3 has AA or … or pl.9 has AA) ~ P(player2 has AA) + P(player3 has AA) + … + P(player9 has AA) = 8 x choose(4,2) / choose(50,2) = 3.9%, or 1 in 26. ----------- What is exactly P(SOMEONE has an Ace, given you have KK)? (8 opponents) (or more than one ace) P(SOMEONE has an Ace) = 100% - P(nobody has an Ace). P(nobody has an Ace) = P(pl2 doesn’t have one & pl.3 doesn’t & … & pl.9 doesn’t) = P(pl.2 doesn’t) x P(pl.3 doesn’t | pl.2 doesn’t) x … x P(pl.9 doesn’t | 2-8 don’t) = choose(46,2)/choose(50,2) x choose(44,2)/choose(50,2) x … x ch(32,2)/ch(50,2) = 1.6%. So P(SOMEONE has an Ace) = 98.4%. River: 7  !

4. 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 or 5A) = (2/45 x 1/44) + (2/45 x 2/44) + (2/45 x 2/44) = 0.505%. 1 in 198. P(Nguyen wins) = P(77) = 3/45 x 2/44 = 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, 3/41 x 2/40 = 1 in 273.3). ] * Szentkuti was eliminated next hand, in 4th place. Nguyen went on to win it all. Turn: 7 . River: 7  !

5. 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%.

6. 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, it seems very reasonable to assume he had a 35.2% chance to win. (Harrington: 10%!) * Turn: A  ! River: 2 . * Hansen was eliminated 2 hands later, in 3rd place. Martens then lost to Nguyen.  

7. LLN CLT, Variance and prop bets. 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. Expected value (µ) = ∑ y P(y) Sample mean (X) = ∑X i / n Sample standard deviation = √[∑(X i - X) 2 / (n-1)] iid: independent and identically distributed. Suppose X 1, X 2, etc. are iid with expected value µ and sd , LAW OF LARGE NUMBERS: X ---> µ. CENTRAL LIMIT THEOREM: (X - µ) ÷ (  /√n) ---> Standard Normal.

9. 95% between -1.96 and 1.96

10. Truth: -49 to 51, exp. value = 1.0

11. Estimated as X +/- 1.96  /√n =.95 +/- 0.28

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