2. Stat 35b: Introduction to Probability with Applications to Poker Outline for the day: 1.hw, terms, etc. 2.WSOP example 3. permutations, and combinations 4.“Addition rule”. 5.Examples involving combinations & addition rule. 6.Kolberg/Murphy example 7.R   u 

3. 3. Permutations and Combinations Basic counting principle: If there are a 1 distinct possible outcomes on experiment #1, and for each of them, there are a 2 distinct possible outcomes on experiment #2, etc., then there are a 1 x a 2 x … x a j 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 10 67

4. A combination is a collection of outcomes, where order doesn’t matter. e.g. in hold’em, how many distinct 2-card hands are possible? 52 x 51 if order matters, but then you’d be double-counting each [ since now (A , K  ) = (K , A  ).] So, the number of distinct hands where order doesn’t matter is 52 x 51 / 2. In general, with n distinct objects, the # of ways to choose k different ones, where order doesn’t matter, is “n choose k” = C(n,k) = ( n ) = n!. k k! (n-k)!

5. 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 u s, and there are exactly 2 u s 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 u s, 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…

6. ADDITION RULE, revisited….. Axioms (initial assumptions/rules) of probability: 1)P(A) ≥ 0. 2)P(A) + P(A c ) = 1. 3)Addition rule: If A 1, A 2, A 3, … are mutually exclusive, then P(A 1 or A 2 or A 3 or … ) = P(A 1 ) + P(A 2 ) + P(A 3 ) + … 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). A B C

7. Ex. You have 2 u s, and there are exactly 2 u s 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 u s, 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%.

8. 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 it’s 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%

9. Example: Poker Royale: Comedians vs. Poker Pros, Fri 9/23/05. Linda Johnson $543,000Kathy Kolberg $300,000 Phil Laak $475,000Sue Murphy $155,000 Tammy Pescatelli $377,000Mark Curry $0. No small blind. Johnson in big blind for $8000. Murphy (8  8  ). Calls $8,000. Kolberg. (9  9 u ). Raises to $38,000. Pescatelli (Kh 3  ) folds, Laak (9  3  ) folds, Johnson (J  6 u ) folds. Murphy calls. TV Screen: Kolberg. (9  9 u ) 81% Murphy (8  8  ) 19% Flop: 8  T u T . Murphy quickly goes all in. Kolberg thinks for 2 min, then calls. Laak (to Murphy): “You’re 92% to take it down.” TV Screen:Kolberg. (9  9 u ) 17% Murphy (8  8  ) 83% Who’s right? (Turn 9  river A u ), so Murphy is eliminated. Laak went on to win.

10. TV Screen: Kolberg. (9  9 u ) 81%Murphy (8  8  ) 19% Flop: 8  T u T . Murphy quickly goes all in. Kolberg thinks for 2 min, then calls. Laak (to Murphy): “You’re 92% to take it down.” TV Screen:Kolberg. (9  9 u ) 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) = 8.8%Given other players’ 6 cards? Laak had a 9, so it’s 1/39 + 1/39 = 5.1%

11. 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% TV Screen: Kolberg. (9  9 u ) 81%Murphy (8  8  ) 19% Flop: 8  T u T . Murphy quickly goes all in. Kolberg thinks for 2 min, then calls. Laak (to Murphy): “You’re 92% to take it down.” TV Screen:Kolberg. (9  9 u ) 17%Murphy (8  8  ) 83% Cardplayer.com: 16.8% 83.2% other players’ 6 cards? Laak had a 9, so it’s 1/39 + 1/39 = 5.1%