Expected value y Py

1
Expected value (µ) ? y P(y)
Sample mean (X) ?Xi / n
Sample standard deviation v?(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 ---gt µ .
CENTRAL LIMIT THEOREM (X - µ) (s/vn)
---gt Standard Normal.
2
95 between -1.96 and 1.96
3
Truth -49 to 51, exp. value 1.0
4
Truth -49 to 51, exp. value 1.0 Estimated as X
/- 1.96 s/vn .95 /- 0.28
5
Poker has high standard deviation.
Important to keep track of results.
Dont just track ?Xi.
Track X /- 1.96 s/vn .
Make sure its converging to something positive.
6
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.
7
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 youre
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, youre 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.
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 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 7s on the turn and river are
2741. 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.
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) (9696) (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, hed 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?!

• Hansen was eliminated 2 hands later, in 3rd
  place. Martens then lost to Nguyen.
• ? ? ?