Stat 35b: Introduction to Probability with Applications to Poker

Outline for the day:

1. Deal-making and expected value

2. Odds ratios, revisited

3. Variance and SD

4. Booth vs. Ivey

Hand in hw2!!!

1. 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.

For a fair deal, the amount you win = the expected value of the amount you will win.

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!

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.

2. Odds ratios, revisited:Odds ratio of A = P(A)/P(Ac)Odds against A = Odds ratio of Ac = P(Ac)/P(A).

An advantage of probability over odds ratios is the mult. rule:P(A & B) = P(A) x P(B|A), but can’t multiply odds ratios.

Example: High Stakes Poker, 1/25/07, on GSN:Blinds $300/$600 + $1200 “straddle” + eight $100 antes = $2900.Gabe Kaplan: K K, raises to $4200. Chen folds, Doyle folds, Matusow calls with A 9.Gold folds, Alaie folds, Sheikhan folds, Ramdin folds.

Flop: T 8 J. Kaplan checks, Matusow bets $8000, Kaplan calls.Turn: A. (both players check)River: 6. Kaplan bets $16000, Matusow folds! (pot $43,300)

Gabe Kaplan: K K. Mike Matusow: A 9.

Flop: T 8 J.

To win, Matusow needs a 7, Q, or A.(But not Q9 or QA, and they tie with 79…. And not two more s. )And, Matusow can also win with two more s.So, part of figuring out P(Matusow winning) would be to get:

Given their cards and the flop, what’s P(Matusow makes a flush)?

P( on turn AND on river) = P( turn) x P( river | turn) = 9/45 x 8/44 = 3.6%.

Odds against it? 96.3% ÷ 3.6% = 26.75 : 1.Odds against on turn = 4 : 1. Odds against on river | on turn

= 4.5 : 1.Multiplying, you’d get (4 x 4.5) = 18. Moral: you can’t multiply odds ratios!

3. Variance and SD For a discrete random variable X,

E(X) = ∑ f(y) x y = µ.For instance, if you roll a die, and you lose $100 (i.e. win $-100) if you roll a 1 or 2,

you win $50 if you roll 3-5,and you win $20 if you roll a 6.

If X = your winnings, then E(X) = $-100 x 2/6 + $50 x 3/6 + $20 x 1/6 = $-5.

If you roll three dice, and you pay $10 to play, but you get back $10 for each 6 you roll,then if X = your profit after playing this game once, thenE(X) = $-10 x P(zero 6s) + $0 x P(one 6) + $10 x P(two 6s) + $20 x P(three 6s) = $-10 x 5/6 x 5/6 x 5/6 + 0 + $10 x 3 x 1/6 x 1/6 x 5/6 + $20 x 1/6 x 1/6 x 1/6 = $-5.

But the two games seem different: the first has higher variance, 2:2 =V(X) = E(X-µ)2 = ∑ f(y) x (y-µ)2 = E(X2) - µ2. = standard deviation = √2. Indicates how far a typical observation is from µ.

For instance, in game 1, E(X2) = ($-100)2 x 2/6 + ($50)2 x 3/6 + ($20)2 x 1/6 = $2150.µ2 = (-$5)2 = $25, so 2 = $2150 - $25 = $2125, and = √$2125 ~ $46.Game 2: E(X2) = ($-10)2 x (5/6)3 + 0 + ($10)2 x 3 x 1/6 x1/6 x 5/6 + ($20)2 x (1/6)3 ~ $66.7.µ2 = (-$5)2 = $25, so 2 = $66.7 - $25 = $41.7, and = √$41.7 ~ $6.50.