1. Stat 35b: Introduction to Probability with Applications to Poker
Outline for the day:
Review list.
Farha vs. Antonius, expected value and variance.
Flush draws and straight draws.
Bayes’s rule example.
5. E(cards til 2nd king) again.
6. Rainbow flops, E(X), and SD(X).
Midterm is Thur, Feb 21, in class. 11:10-12: min.
Open book plus one page of notes, double sided. Bring a calculator!
  u    u 

2. 1. Review List
Basic principles of counting.
Axioms of probability, and addition rule.
Permutations & combinations.
Conditional probability.
Independence.
Multiplication rules. P(AB) = P(A) P(B|A) [= P(A)P(B) if ind.]
Odds ratios.
Random variables (RVs).
Discrete RVs, and probability mass function (pmf).
Expected value.
Pot odds calculations.
Luck, skill, and deal-making.
Variance and SD.
Bernoulli RV. [ µ = p, s = √(pq). ]
Binomial RV. [# of successes, out of n tries. µ = np, s = √(npq).]
Geometric RV. [# of tries til 1st success. µ = 1/p, s = (√q) / p. ]
Negative binomial RV. [# of tries til rth success. µ = r/p, s = (√rq) / p. ]
E(X+Y), V(X+Y) (ch. 7.1).
Bayes’s rule (ch. 3.4).
Basically, we’ve done all of ch. 1-5 except 4.6, 4.7, and 5.5.
We’ve also done most of but this won’t be on the midterm.

3. 2) Farha vs. Antonius, expected value and variance.
Recall that E(X+Y) = E(X) + E(Y). Whether X & Y are independent or not!
Similarly, E(X + Y + Z + …) = E(X) + E(Y) + E(Z) + …
And, if X & Y are independent, then V(X+Y) = V(X) + V(Y).
so SD(X+Y) = √[SD(X)^2 + SD(Y)^2].
Also, if Y = 9X, then E(Y) = 9E(Y), and SD(Y) = 9SD(X). V(Y) = 81V(X).
Farha vs. Antonius.
Running it 4 times. Let X = chips you have after the hand. Let p be the prob. you win.
X = X1 + X2 + X3 + X4, where X1 = chips won from the first “run”, etc.
E(X) = E(X1) + E(X2) + E(X3) + E(X4)
= 1/4 pot (p) + 1/4 pot (p) + 1/4 pot (p) + 1/4 pot (p)
= pot (p)
= same as E(Y), where Y = chips you have after the hand if you ran it once!
But the SD is smaller: clearly X1 = Y/4, so SD(X1) = SD(Y)/4. So, V(X1) = V(Y)/16.
V(X) ~ V(X1) + V(X2) + V(X3) + V(X4),
= 4 V(X1)
= 4 V(Y) / 16
= V(Y) / So SD(X) = SD(Y) / 2.

4. 3) Flush draws & Straight draws.
Probability needed to call all-in: P(win) must be ≥ Bet ÷ (Pot + Bet).
If your opponent bets the amount that was previously in the pot, then this probability is 1/3.
Suppose you have two s & there are exactly two s on the flop. [No info about opponents.]
Given this info, P(making a flush)?
= P(at least one more  on turn or river)
= 1 - P(non- on turn AND non- on river)
= 1 - choose(38,2) ÷ choose(47,2)
= 35.0%.
*** However, this assumes you’ll get to see both the turn and the river!
P(making an open-ended straight draw on turn or river)? [8 outs.]
= 1 - P(non-out on turn AND non-out on river)
= 1 - choose(39,2) ÷ choose(47,2)
= 31.5%.
*** However, if you hit your straight and bet, your opponent might call!
Implied odds: P(win) must be ≥ Bet ÷ (Pot + Bet + extra amount you’ll win later)

5. 4. Bayes’ rule example.
Suppose P(your opponent has the nuts) = 1%, and P(opponent has a weak hand) = 10%.
Your opponent makes a huge bet. Suppose she’d only do that with the nuts or a weak hand, and that P(huge bet | nuts) = 100%, and P(huge bet | weak hand) = 30%.
What is P(nuts | huge bet)?
P(nuts | huge bet) =
P(huge bet | nuts) * P(nuts)
P(huge bet | nuts) P(nuts) + P(huge bet | horrible hand) P(horrible hand)
= % * 1%
100% * 1% % * 10%
= 25%.

6. 5. E(cards til 2nd king), revised.
Z = the number of cards til the 2nd king. What is E(Z)?
Let X1 = number of non-king cards before 1st king.
Let X2 = number of non-kings after 1st king til 2nd king.
Let X3 = number of non-kings after 2nd king til 3rd king.
Let X4 = number of non-kings after 3rd king til 4th king.
Let X5 = number of non-kings after 4th king til the end of the deck.
Clearly, X1 + X2 + X3 + X4 + X5 + 4 = 52.
By symmetry, E(X1) = E(X2) = E(X3) = E(X4) = E(X5).
Therefore, E(X1) = E(X2) = 48/5.
Z = X1 + X2 + 2, so E(Z) = E(X1) + E(X2) = 48/5 + 48/5 + 2 = 21.2.

7. 6. Rainbow flops.
P(Rainbow flop) = choose(4,3) * * 13 * ÷ choose(52,3)
choices for the 3 suits numbers on the 3 cards possible flops
~ 39.76%.
Q: Out of 100 hands, what is the expected number of rainbow flops? +/- what?
X = Binomial (n,p), with n = 100, p = 39.76%, q = 60.24%.
E(X) = np = 100 * = 39.76
SD(X) = √(npq) = sqrt(23.95) = 4.89.
So, expect around / rainbow flops, out of 100 hands.

8. 6. Rainbow flops, continued.
P(Rainbow flop) ~ 39.76%.
Q: Let X = the number of hands til your 4th rainbow flop.
What is P(X = 10)? What is E(X)? What is SD(X)?
X = negative binomial (r,p), with r = 4, p = 39.76%, q = 60.24%.
P(X = k) = choose(k-1, r-1) pr qk-r.
Here k = 10. P(X = 10) = choose(9,3) 39.76% %6 = 10.03%.
µ = E(X) = r/p = 4 ÷ = hands.
s = SD(X) = (√rq) / p = sqrt(4*0.6024) / = 3.90 hands.
So, you expect it typically to take around
/ hands til your 4th rainbow flop.