1. Stat 35b: Introduction to Probability with Applications to Poker
Outline for the day:
Turn in Hw2. Hw3 assigned.
Expected value and pot odds, continued
Violette/Elezra example
Yang / Kravchenko
Deal-making and expected value
Reminder: project A is due by by Wed, Feb 8, 8pm.
You may me for your teammate’s address.
  u    u 

2. Turn in Hw2.
Hw3 is due Thur Mar 1, 12:30pm.
4.7, 4.8, 4.12, 4.16, 5.6, 6.2.
Also, read chapter 5.
Reminder: project A is due by by Wed, Feb 8, 8pm.
You may me for your teammate’s address.
Just submit one per team.
  u    u 

3. 2) Pot odds and expected value, continued.
From a previous lecture: to call an all-in, need P(win) > B ÷ (B+pot).
Expressed as an odds ratio, this is sometimes referred to as pot odds or express odds.
If the bet is not all-in & another betting round is still to come, need
P(win) > wager ÷ (wager + winnings),
where winnings = pot + amount you’ll win on later betting rounds,
wager = total amount you will wager including the current round & later rounds,
assuming no folding.
The terms Implied-odds / Reverse-implied-odds describe the cases where
winnings > pot or where wager > B, respectively. See p66.

4. 3) Example: Poker Superstars Invitational Tournament, FSN, October 2005.
Ted Forrest: 1 million chips
Freddy Deeb: 825,000 Blinds: 15,000 / 30,000
Cindy Violette: 650,000
Eli Elezra: 575,000
* Elezra raises to 100,000 
* Forrest folds.
* Deeb, the small blind, folds.
* Violette, the big blind with Ku Ju, calls.
* The flop is: 2u 7 Au
* Violette bets 100,000.
* Elezra raises all-in to 475,000. (pot = 790,000)
So, it's 375,000 more to Violette. She folds.
Q: Based on expected value, should she have called?
Her chances must be at least 375,000 / (790, ,000) = 32%.3

5. Violette has Ku Ju. The flop is: 2u 7 Au.
Q: Based on expected value, should she have called?
Her chances must be at least 375,000 / (790, ,000) = 32%.
vs. AQ: 38% AK: 37% AA: 26% : 26% A7: 31%
A2: 34% : 34% TT: 54% T9: 87% 73: 50%
Harrington's principle: always assume at least a 10% chance that opponent is bluffing.
Bayesian approach: average all possibilities, weighting them by their likelihood.
Maybe she's conservative.... but then why play the hand at all?
Reality: Elezra had 7u 3. Her chances were 51%. Bad fold.
What was her prob. of winning (given just her cards and Elezra’s, and the flop)?
Of choose(45,2) = 990 combinations for the turn & river, how many give her the win?
First, how many outs did she have? eight us + 3 kings + 3 jacks = 14.
She wins with (out, out) or (out, nonout) or (non-u Q, non-u T)
choose(14,2) x * 3 = 534
but not (k or j, 7 or non-u 3) and not (3u , 7 or non-u 3)
- 6 * * 4 = 506.
So the answer is 506 / 990 = 51.1%.

6. 4. Yang / Kravchenko.
Yang A 10u. Pot is 19million. Bet is 8.55 million.
Needs P(win) > 8.55 ÷ ( ) = 31%.
vs. AA: 8.5%. AJ-AK: 25-27%. KK-TT: 29% : 44-48%. KQs: 56%.
Bayesian method: average these probabilities, weighting each by its likelihood.
See p49-53.

7. 4. Yang / Kravchenko.
Yang A 10u. Pot is 19.0 million. Bet is 8.55 million.
Suppose that, averaging the different probabilities, P(Yang wins) = 30%.
And say Yang calls. Let X = the number of chips Kravchenko has after the hand.
What is E(X)? [Note, if Yang folds, then X = 19.0 million for sure.]
E(X) = ∑ [k * P(X=k)]
= [0 * 30%] + [27.55 million * 70%]
= million.

8. 5. Deal-making. (Expected value, game theory)
Game-theory: For a symmetric-game tournament, the probability of winning is approx. optimized by the myopic rule (in each hand, maximize your expected number of chips),
and
P(you win) = your proportion of chips (Theorems and on pp ).
For a fair deal, the amount you win = the expected value of the amount you will win. See p61.

9. For instance, suppose a tournament is winner-take-all, for $8600.
With 6 players left, you have 1/4 of the chips left.
An EVEN SPLIT would give you $8600 ÷ 6 = $1433.
A PROPORTIONAL SPLIT would give you $8600 x (your fraction of chips)
= $8600 x (1/4) = $2150.
A FAIR DEAL would give you the expected value of the amount you will win
= $8600 x P(you get 1st place) = $2150.
But suppose the tournament is not winner-take-all, but pays
$3800 for 1st, $2000 for 2nd, $1200 for 3rd, $700 for 4th, $500 for 5th, $400 for 6th.
Then a FAIR DEAL would give you
$3800 x P(1st place) + $2000 x P(2nd) +$1200 x P(3rd)+$700xP(4th) +$500xP(5th) +$400xP(6th).
Hard to determine these probabilities. But, P(1st) = 25%, and you might roughly estimate the others as P(2nd) ~ 20%, P(3rd) ~ 20%, P(4th) ~ 15%, P(5th) ~10%, P(6th) ~ 10%, and get
$3800 x 25% + $2000 x 25% +$1200 x 20% + $700x 15% + $500x 10% +$400x 5% = $1865.
If you have 40% of the chips in play, then:
EVEN SPLIT = $1433.
PROPORTIONAL SPLIT = $3440.
FAIR DEAL ~ $2500!

10. Another example. Before the Wasicka/Binger/Gold hand,
Gold had 60M, Wasicka 18M, Binger 11M.
Payouts: 1st place $12M, 2nd place $6.1M, 3rd place $4.1M.
Proportional split: of the total prize pool left, you get your proportion of chips in play.
e.g. $22.2M left, so Gold gets 60M/(60M+18M+11M) x $22.2M ~ $15.0M.
A fair deal would give you
P(you get 1st place) x $12M P(you get 2nd place) x $6.1M + P(3rd pl.) x $4.1M .
*Even split: Gold $7.4M, Wasicka $7.4M, Binger $7.4M.
*Proportional split: Gold $15.0M, Wasicka $4.5M, Binger $2.7M.
*Fair split: Gold $10M, Wasicka $6.5M, Binger $5.7M.
*End result: Gold $12M, Wasicka $6.1M, Binger $4.1M.

11. Luck vs. skill. pp 71-79. Any thoughts?
name1 = c("gravity","tommy","ursula","timemachine","vera","william","xena”)
decision1 = list(gravity, tommy, ursula, timemachine, vera, william, xena)
tourn1(name1, decision1, myfast1 = 2) ## run quickly
tourn1(name1, decision1, myfast1 = 0) ## run slowly, showing key hands