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 s ,. LAW OF LARGE NUMBERS : X ---> µ .
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Sample mean (X) = ∑Xi / n
Sample standard deviation = √[∑(Xi - X)2 / (n-1)]
iid: independent and identically distributed.
Suppose X1, X2 , etc. are iid with expected value µ and sd s ,
LAW OF LARGE NUMBERS:
X ---> µ .
CENTRAL LIMIT THEOREM:
(X - µ) ÷ (s/√n) ---> Standard Normal.
Truth: -49 to 51, exp. value = 1.0
Estimated as X +/- 1.96 s/√n = .95 +/- 0.28
* Poker has high standard deviation.
Important to keep track of results.
* Don’t just track ∑Xi.
Track X +/- 1.96 s/√n .
Make sure it’s converging to something positive.
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 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.
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.
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). ]
Turn: 7.
River: 7!
* Szentkuti was eliminated next hand, in 4th place. Nguyen went on to win it all.
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 101 Shooting Star. 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%!)
River: 2.
* Turn: A!