1. Stat 35b: Introduction to Probability with Applications to Poker
Outline for the day:
Multiple callers with KK
Pot odds calculations
Key hand from the 2012 main event.
Wasicka/Gold/Binger example
P(flop a full house).
HW2 is due Thur, Jan 31, 11am. 2.6, 2.9, 2.10, 2.18, 3.2, and 3.6.
Next time: Uniform random variables in R, Project A teams
  u    u 

2. Want multiple callers with KK?
a) You have $100 and KK and are all-in against TT. You're 81% to double up, so your expected number of chips after the hand is
0.81 x $200 = $162.
b) You have $100 and KK and are all-in against A9 and TT. You're 58% to have $300, so your expected value is $174.
So, if you have KK and an opponent with TT has already called you, and another who has A9 is thinking about whether to call you too, you actually want A9 to call!
Given this, one may question Harrington's suggested strategy of raising huge in order to isolate yourself against one player.

3. 2. POT ODDS CALCULATIONS.
Suppose someone bets (or raises) you, going all-in. What should your chances of winning be in order for you to correctly call?
Let B = the amount bet to you, i.e. the additional amount you'd need to put in if you want to call. So, if you bet 100 & your opponent with 800 left went all-in, B = 700.
Let POT = the amount in the pot right now (including your opponent's bet).
Let p = your probability of winning the hand if you call. So prob. of losing = 1-p.
Let CHIPS = the number of chips you have right now.
If you call, then E[your chips at end] = (CHIPS - B)(1-p) + (CHIPS + POT)(p)
= CHIPS(1-p+p) - B(1-p) + POT(p) = CHIPS - B + Bp + POTp
If you fold, then E[your chips at end] = CHIPS.
You want your expected number of chips to be maximized, so it's worth calling if
-B + Bp + POTp > 0, i.e. if p > B / (B+POT).3/39 + 3/39 - C(3,2)/C(39,2) = 15.0%

4. 4. Example: 2006 World Series of Poker (WSOP).   u 
Blinds: 200,000/400,000, + 50,000 ante.
Jamie Gold (4 3): 60 million chips. Calls.
Paul Wasicka (8 7): 18 million chips. Calls.
Michael Binger (A 10): 11 million chips. Raises to $1,500,000.
Gold & Wasicka call (pot = 4,650,000) Flop: 6 10 5.
Wasicka checks, Binger bets $3,500,000. (pot = 8,150,000)
Gold moves all-in for 16,450,000. (pot = 24,600,000)
Wasicka folds. Q: Based on expected value, should he have called?
If Binger will fold, then Wasicka’s chances to beat Gold must be at least
16,450,000 / (24,600, ,450,000) = 40.1%.
If Binger calls, it’s a bit complicated, but basically Wasicka’s chances must be at least 16,450,000 / (24,600, ,450, ,950,000) = 35.0%.

5. 4. Example: 2006 World Series of Poker (WSOP), continued
Jamie Gold (4 3). Paul Wasicka (8 7). Michael Binger (A 10).
Flop: 6 10 5.
Given only Wasicka’s cards and the flop,
P( or 9 or 4 on turn or river, or 77 or 88)
= P(out & out) + P(out & non-out) + P(77) + P(88) [all 15 s,9s,4s are all outs]
= (choose(15,2) + 15 * 32 + choose(3,2) + choose(3,2)) / choose(47,2)
= 54.7%.
Worst case scenario: what if only non- 9s or 4s, or 77 or 88 are outs?
P(out & out) + P(out & non--non-out) + P(77) + P(88) [non- 9s & 4s are outs]
= (choose(6,2) + 6 * 32 + choose(3,2) + choose(3,2)) / choose(47,2)
= 19.7%.

6. 5. Suppose you’re all in next hand, no matter what cards you get.
P(flop a full house)?
Key idea: forget order! Consider all combinations of your 2 cards and the flop.
Sets of 5 cards. Any such combo is equally likely! choose(52,5) different ones.
P(flop full house) = # of different full houses / choose(52,5)
How many different full houses are possible?
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.