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
• hw, terms, etc.
• WSOP example
• 3. permutations, and combinations
• Addition rule.
• Examples involving combinations addition rule.
• Kolberg/Murphy example
• R

? ? u ? ? ? u ?
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3. Permutations and Combinations
Basic counting principle If there are a1
distinct possible outcomes on experiment 1, and
for each of them, there are a2 distinct possible
outcomes on experiment 2, etc., then there are
a1 x a2 x x aj distinct possible ordered
outcomes on the j experiments. e.g. you get 1
card, opp. gets 1 card. of distinct
possibilities? 52 x 51. ordered (A? , K?) ?
(K? , A?) . Each such outcome, where order
matters, is called a permutation. Number of
permutations of the deck? 52 x 51 x x 1
52! 8.1 x 1067
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A combination is a collection of outcomes, where
order doesnt matter. e.g. in holdem, how many
distinct 2-card hands are possible? 52 x 51 if
order matters, but then youd be double-counting
each since now (A? , K?) (K? , A?)
. So, the number of distinct hands where order
doesnt matter is 52 x 51 / 2. In general,
with n distinct objects, the of ways to choose
k different ones, where order doesnt matter,
is n choose k C(n,k) (n) n!
.
k k! (n-k)!
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k! 1 x 2 x x k. (convention 0! 1. )
(n choose k) C(n,k) (n) n!
. k
k! (n-k)! Ex. You have 2 us, and there are
exactly 2 us on the flop. Given this info, what
is P(at least one more u on turn or river)?
Answer 52-5 47 cards left (9 us, 38
others). So n C(47,2) 1081 combinations for
next 2 cards. Each equally likely (and obviously
mutually exclusive). Two-u combos C(9,2) 36.
One-u combos 9 x 38 342. Total 378. So
answer is 378/1081 35.0. ----------------------
-------------------------------- Answer 2 Use
the addition rule
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• ADDITION RULE, revisited..
• Axioms (initial assumptions/rules) of
  probability
• P(A) 0.
• P(A) P(Ac) 1.
• Addition rule
• If A1, A2, A3, are mutually exclusive,
  then
• P(A1 or A2 or A3 or ) P(A1) P(A2)
  P(A3)

A
B
C
As a result, even if A and B might not be
mutually exclusive, P(A or B) P(A)
P(B) - P(A and B).
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Ex. You have 2 us, and there are exactly 2 us on
the flop. Given this info, what is P(at least one
more u on turn or river)? Answer 52-5 47
cards left (9 us, 38 others). So n C(47,2)
1081 combinations for next 2 cards. Each equally
likely (and obviously mutually exclusive).
Two-u combos C(9,2) 36. One-u combos 9 x 38
342. Total 378. So answer is 378/1081
35.0. -------------------------------------------
----------- Answer 2 Use the addition
rule. P( 1 more u) P(u on turn OR river)
P(u on turn) P(u on river) - P(both)
9/47 9/47 - C(9,2)/C(47,2) 19.15
19.15 - 3.3 35.0.
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Ex. You have AK. Given this, what is P(at least
one A or K comes on board of 5 cards)? Wrong
Answer P(A or K on 1st card) P(A or K on 2nd
card) 6/50 x 5 60.0. No these
events are NOT Mutually Exclusive!!! Right
Answer C(50,5) 2,118,760 boards possible. How
many have exactly one A or K? 6 x C(44,4)
814,506 How many have exactly 2 aces or kings?
C(6,2) x C(44,3) 198,660 How many have exactly
3 aces or kings? C(6,3) x C(44,2) 18,920
altogether, 1032752 boards have at least one A or
K, So its 1032752 / 2118760 48.7. Easier
way P(no A or K) C(44,5)/C(50,5) 1086008 /
2118760 51.3, so answer 100 - 51.3 48.7
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Example Poker Royale Comedians vs. Poker
Pros, Fri 9/23/05. Linda Johnson
543,000 Kathy Kolberg 300,000 Phil Laak
475,000 Sue Murphy 155,000 Tammy
Pescatelli 377,000 Mark Curry 0.
No small blind. Johnson in big blind for
8000. Murphy (8? 8?). Calls 8,000. Kolberg. (9?
9u). Raises to 38,000. Pescatelli (Kh 3?)
folds, Laak (9? 3?) folds, Johnson (J? 6u)
folds. Murphy calls. TV Screen Kolberg. (9?
9u) 81 Murphy (8? 8?) 19 Flop 8? Tu
T?. Murphy quickly goes all in. Kolberg thinks
for 2 min, then calls. Laak (to Murphy) Youre
92 to take it down. TV Screen Kolberg. (9? 9u)
17 Murphy (8? 8?) 83 Whos right? (Turn 9?
river Au), so Murphy is eliminated. Laak went on
to win.
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TV Screen Kolberg. (9? 9u) 81 Murphy (8? 8?)
19 Flop 8? Tu T?. Murphy quickly goes all
in. Kolberg thinks for 2 min, then calls. Laak
(to Murphy) Youre 92 to take it down. TV
Screen Kolberg. (9? 9u) 17 Murphy (8? 8?)
83 Cardplayer.com 16.8 83.2 Laak
(about Kolberg) She has two outs twice. P(9 on
the turn or river, given just their 2 hands and
the flop)? P(9 on turn) P(9 on river) - P(9
on both) 2/45 2/45 - 1/C(45,2) 8.8Given
other players 6 cards? Laak had a 9, so its
1/39 1/39 5.1
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TV Screen Kolberg. (9? 9u) 81 Murphy (8? 8?)
19 Flop 8? Tu T?. Murphy quickly goes all
in. Kolberg thinks for 2 min, then calls. Laak
(to Murphy) Youre 92 to take it down. TV
Screen Kolberg. (9? 9u) 17 Murphy (8? 8?)
83 Cardplayer.com 16.8
83.2 other players 6 cards? Laak had a 9, so
its 1/39 1/39 5.1
Given just their 2 hands and the flop, what
is P(9 or T on the turn or river)? P(9 or T on
the turn) P(9 or T on river) - P(both) 4/45
4/45 - C(4,2)/C(45,2) 17.2