Stat 35b: Introduction to Probability with Applications to Poker

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• Stat 35b Introduction to Probability with
  Applications to Poker
• Outline for the day
• Collect hw3, give out hw4, no class Thur Oct 29!
• Play (and lose) like the pros
• Savage / Tyler
• P(flop a full house)
• P(Rainbow flop)
• Szenkuti / Nguyen
• Remember project A code due Mon 8pm by email!

? ? u ? ? ? u ?
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2. Play (and lose) like the pros. Phil Hellmuth,
Play Poker Like the Pros, Collins, 2003. Strategy
for beginners AA, KK, QQ, or AK. P(getting one
of these hands)? 3 x choose(4,2)/choose(52,2)
4x4/choose(52,2) 3 x 6/1326 16/1326 3
x 0.45 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.
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3. Savage / Tyler. 4. P(flop a full house)? (If
youre all in next hand, no matter what cards you
get. Key idea forget order! Consider all
combinations of your 2 cards and the
flop. P(flop full house) of different full
houses / choose(52,5) 13 choose(4,3)
different choices for the triple. For each such
choice, there are 12 choose(4,2) choices left
for the pair. So, P(flop full house) 13
choose(4,3) 12 choose(4,2) / choose(52,5)
0.144, or 1 in 694. Notice the difference
between this calculation and the probability of
flopping 2 pairs. Here, its 13 12
possibilities for the number on the triplet and
the pair instead of choose(13,2) possibilities
for the numbers on the two pairs.
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5. Rainbow board all different suits. P(rainbow
flop)? How many ways can this happen? Pick 3
suits. For each, 13 possible cards. 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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6. Nguyen / Szenkuti. 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) (1 22) choose(45,2)
0.505. 1 in 198. P(Nguyen wins) P(77)
choose(3,2) choose(45,2) 0.30. 1 in
330. Note tv said odds of running 7s on the
turn and river are 2741. Given
Hansen/Martens cards, choose(3,2) choose(41,2)
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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7. Martens / Hansen. 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,
wrong move if he knew what Hansen had. Turn A?!
River 2?. Hansen was eliminated 2 hands later,
in 3rd place. Martens then lost to Nguyen.