Stat 35b: Introduction to Probability with Applications to Poker

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• Stat 35b Introduction to Probability with
  Applications to Poker
• Poker Code competition all-in or fold.

? ? u ? ? ? u ?
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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 doesnt have one pl.3 doesnt pl.9
doesnt) P(pl.2 doesnt) x P(pl.3 doesnt
pl.2 doesnt) x x P(pl.9 doesnt 2-8 dont)
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?!
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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.
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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.
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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.
• ? ? ?

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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.
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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.
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95 between -1.96 and 1.96
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Truth -49 to 51, exp. value 1.0
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Truth -49 to 51, exp. value 1.0 Estimated as X
/- 1.96 s/vn .95 /- 0.28
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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.