Correction'

1

• Correction.
• Meaning of Probability
• 3. Axioms of Probability
• Addition rule
• Multiplication rules
• Examples

• Correction from end of last time
• of possible 2-card hands choose(52,2) 1326,
  not 221.
• P(A?A?) 1/1326.
• P(AA) Choose(4,2)/1326 1/221.

2

• Notation P(A) 60. A is an event.
• Not P(60).
• Meaning of probability
• Frequentist If repeated independently under
  the same conditions millions and millions of
  times, A would happen 60 of the times.
• Bayesian Subjective feeling about how likely
  something seems.
• P(A or B) means P(A or B or both )
• Mutually exclusive P(A and B) 0.
• Independent P(A given B) written P(AB)
  P(A).
• P(Ac) means P(not A).

3

• Axioms (initial assumptions/rules) of
  probability
• P(A) 0.
• P(A) P(Ac) 1.
• If A1, A2, A3, are mutually exclusive, then
• P(A1 or A2 or A3 or ) P(A1) P(A2)
  P(A3)
• (3 is sometimes called the addition
  rule)
• Probability ltgt Area. Measure theory, Venn
  diagrams

B
A
P(A or B) P(A) P(B) - P(A and B).
4
A
B
C
Fact P(A or B) P(A) P(B) - P(A and
B). P(A or B or C) P(A)P(B)P(C)-P(AB)-P(AC)-P(
BC)P(ABC).
Fact If A1, A2, , An are equally likely
mutually exclusive, and if P(A1 or A2 or or
An) 1, then P(Ak) 1/n. So, you can
count P(A1 or A2 or or Ak) k/n. Ex. You
have 76, and the board is KQ54.
P(straight)? 52-2-446. P(straight) P(8 on
river OR 3 on river) P(8 on
river) P(3 on river) 4/46 4/46.
5
Counting k! 1 x 2 x x k. ( 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 choices for next 2
cards. Each equally likely (and obviously
mutually exclusive). Choices with a u C(9,2)
9 x 38 378. So answer is 378/1081
35.0. -------------------------------------------
----------- Answer 2 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.
6
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. But 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
7
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 (8h 8s). Calls 8,000. Kolberg. (9c
9d). Raises to 38,000. Pescatelli (Kh 3s)
folds, Laak (9h 3h) folds, Johnson (Jh 6d)
folds. Murphy calls. TV Screen Kolberg. (9c
9d) 81 Murphy (8h 8s) 19 Flop 8c Td
Ts. Murphy quickly goes all in. Kolberg thinks
for 2 min, then calls. Laak (to Murphy) Youre
92 to take it down. TV Screen Kolberg. (9c 9d)
17 Murphy (8h 8s) 83 Whos right? (Turn 9s
river Ad), so Murphy is eliminated. Laak went on
to win.
8
TV Screen Kolberg. (9c 9d) 81 Murphy (8h
8s) 19 Flop 8c Td Ts. Murphy quickly goes
all in. Kolberg thinks for 2 min, then calls.
Laak (to Murphy) Youre 92 to take it
down. TV Screen Kolberg. (9c 9d) 17 Murphy
(8h 8s) 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
9
TV Screen Kolberg. (9c 9d) 81 Murphy (8h
8s) 19 Flop 8c Td Ts. Murphy quickly goes
all in. Kolberg thinks for 2 min, then calls.
Laak (to Murphy) Youre 92 to take it
down. TV Screen Kolberg. (9c 9d) 17 Murphy
(8h 8s) 83 Cardplayer.com 16.8
83.2
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
10
P(A B) is written P(AB). P(A U B) means
P(A or B). Conditional Probability P(A given
B) writtenP(AB) P(AB) / P(B). Independent
A and B are independent if P(AB)
P(A). Fact (multiplication rule for independent
events) If A and B are independent, then P(AB)
P(A) x P(B) Fact (general multiplication
rule) P(AB) P(A) P(BA) P(ABC) P(A)
x P(BA) x P(CAB)
11
Example High Stakes Poker, 1/8/07 (Game Show
Network, Mon/Thur nights) Greenstein folds, Todd
Brunson folds, Harman folds. Elezra calls 600.
Farha (K? J?) raises to 2600 Sheikhan folds.
Negreanu calls, Elezra calls. Pot is
8,800. Flop 6? T? 8?. Negreanu bets 5000.
Elezra raises to 15000. Farha folds. Negreanu
thinks for 2 minutes.. then goes all-in for
another 96,000. Elezra 8? 6?. (Elezra calls.
Pot is 214,800.) Negreanu A? T?. ---------------
----------------------------------------- At this
point, the odds on tv show 73 for Elezra and 25
for Negreanu. They run it twice. First 2? 4?.
Second time?
A?
8?!
P(Negreanu hits an A or T on turn still loses)?
12
Given both their hands, and the flop, and the
first run, what is P(Negreanu hits an A or T on
the turn loses)?
Its P(A or T on turn) x P(Negreanu loses A or
T on the turn) 5/43 x 4/42
1.11 (1 in 90) Note this is very different
from P(A or T on turn) x P(Negreanu loses),
which would be about 5/43 x 73 8.49 (1 in
12)