2. 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.

3. 95% between -1.96 and 1.96

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

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

6. * 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.

7. Phil Helmuth, “Play Poker Like the Pros”, Collins, 2003. strategy for beginners: AA, KK, QQ, or AK. P(getting one of these hands)? 3(4/52)(3/51) + 2/13(4/51) = 1.36% + 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.

8. http://www.freepokerstrategy.com “Unbeatable Texas Holdem Strategy”: all in with AK-AT or pair. P(getting such a hand) = 4 x [16/(52 choose 2)] + 13 x [6/(52 chs 2)] = 4 x 1.2% + 13 x 0.45% = 10.7%. Play 100 times. Expect ~ 11 hands. Pay ~11 blinds = $55. Say you’re called by 88-AA, and AK, for $100 on avg. P(player 1 has one of these) = 7 x 0.45% + 1.2% = 4.4%. P(someone of 8 has one of these) = 1 - (95.6%) 8 = 30%. So, you win pre-flop 70% of the time. (Say $10 on avg.) = 11 x 70% x $10 = $77 profit. Other 30%, you’re on avg about a 65-35 underdog, so you win 11 x 30% x 35% x $100 = $115.50 lose 11 x 30% x 65% x $100 = $214.50. Total: exp. to win $77 + $115.50 - $55 - $214.50 = -$77/ 100 hands.

9. 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  !

10. 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%.

11. 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.  