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.
Rainbow flops, E(X), and SD(X).
Lucky poker.
Project B functions and example.
Midterm is Thur, Feb 23, in class. 75 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. 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.

7. 5. 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.

8. 6. “Lucky Poker”
A, B, C. A & B are a team.
No strategy: only muck (fold) at the end or don’t muck.
First player to two points wins the game.
P(C wins)?
If A has a point and B doesn’t, A should show first. B can muck.
P(C wins in 2 hands) = 1/9 = 3/27
P(C wins in 3 hands) = 4/ (ACC, BCC, CAC, CBC)
P(C wins in 4 hands) = 1/27. Why?
6 ways for this to happen. {CABC, CBAC, ACBC, BCAC, ABCC, or BACC}.
If A has a pt and B doesn’t, then P(B winning a pt on next hand) = 1/6:
{A>B>C, A>C>B, B>A>C, B>C>A, C>A>B, C>B>A}.
So, P(C wins in 4 hands) = 6 x {1/3 x 1/3 x 1/6 x 1/3} = 1/27.
So, P(C wins) = 3/27 + 4/27 + 1/27 = 8/27.

9. 7) Proj. B functions and example.
FUNCTIONS FOR PROJECT B:
straightdraw1 = function(x)
## returns 4 is there are 2 possibilities for a straight.
## returns 2 for a gutshot straight draw.
## returns 0 otherwise
## Note: returns 26 if you already have a straight!
flushdraw1 = function(x)
## returns the max number of one suit
## (4 if flush draw, 5 if a flush already!)
handeval = function(num1,suit1){
Straight-flush: return 8 million - 8,999,999
4 of a kind: return 7 million - 7,999,999
Full house: 6 million - 6,999,999, etc.
. nada 1pr pr kind straight flush full-house 4-kind str-flush .
mil 2mil 3mil 4mil mil mil mil mil mil

10. 7) Proj. B example.
zelda = function(numattable1, crds1, board1, round1, currentbet, mychips1, pot1,
roundbets, blinds1, chips1, ind1, dealer1, tablesleft){
a1 = 0 ## how much I'm gonna end up betting. Note that the default is zero.
a2 = min(mychips1, currentbet) ## how much it costs to call
if(round1 == 1){ ## pre-flop:
## AK: Make a big raise if nobody has yet. Otherwise call.
## AQ: call a small raise, or make one if nobody has yet.
## AJ, AT, KQ, KJ, QJ: call a tiny raise.
## A9, KT, K9, QT, JT, T9: call a tiny raise if in late position (within 2 of the dealer).
## Suited A2-AJ: call a small raise.
## 22-99: call a small raise.
## TT-KK: make a huge raise. If someone's raised huge already, then go all in.
## AA: make a small raise. If there's been a raise already, then double how much it is to you.
a3 = 2*blinds1+1 ## how much a tiny raise would be
a4 = 4*blinds1+1 ## how much a small raise would be
a5 = max(8*blinds1,mychips1/4)+1 ## how much a big raise would be
a6 = max(12*blinds1,mychips1/2)+1 ## how much a huge raise would be
a7 = dealer1 - ind1
if(a7 < -.5) a7 = a7 + numattable1 ## your position: a7 = how many hands til you're dealer
if((crds1[1,1] == 14) && (crds1[2,1] == 13)){ a1 = max(a2,a5) }

11. if((crds1[1,1] == 14) && (crds1[2,1] == 12)){
if(a2 < a4){
a1 = a4
} else if(a2 > a5){
a1 = 0
} else a1 = a2 }
if(((crds1[1,1] == 14) && ((crds1[2,1] < 11.5) && (crds1[2,1] > 9.5))) ||
((crds1[1,1] == 13) && (crds1[2,1] > 10.5)) ||
((crds1[1,1] == 12) && (crds1[2,1] == 11))){
if(a2 < a3) a1 = a2
if(((crds1[1,1] == 14) && (crds1[2,1] == 9)) ||
((crds1[1,1] == 13) && ((crds1[2,1] == 10) || (crds1[2,1] == 9))) ||
((crds1[1,1] == 12) && (crds1[2,1] == 10)) ||
((crds1[1,1] == 11) && (crds1[2,1] == 10)) ||
((crds1[1,1] == 10) && (crds1[2,2] == 9))){
if((a2 < a3) && (a7<2.5)) a1 = a2
if((crds1[1,2] == crds1[2,2]) && (crds1[1,1] == 14) && (crds1[2,1] < 11.5)){
if(a2<a4) a1 = a2
## Note: this trumps the previous section, since it comes later in the code.

12. if((crds1[1,1] == crds1[2,1])){ ## pairs:
if(a2 < a4) a1 = a2
} else if(crds1[1,1] < 13.5){
if(a2<a5) a1 = a5 else a1 = mychips1
} else {
if(a2 < blinds1 + .5) a1 = a4 else a1 = min(2*a2,mychips1)
}}}
if(round1 == 2){ ## post-flop:
## If there's a pair on the board and you don't have a set, then check/call up to small bet.
## Same thing if there's 3-of-a-kind on the board and you don't have a full house or more.
## If you have top pair or an overpair or two pairs or a set, make a big bet (call any bigger bet).
## Otherwise, if nobody's made even a small bet yet, then with prob. 20% make a big bluff bet.
## If you're the last to decide and nobody's bet yet, then increase this prob. to 50%.
## If you have an inside straight draw or flush draw then make a small bet (call any bigger bet).
## If you have a straight or better, then just call.
## Otherwise fold.
a5 = min(sum(roundbets[,1]),mychips1) ## how much big bet would be (prev round's pot size)
a6 = min(.5*sum(roundbets[,1]),mychips1) ## how much a small bet would be
x = handeval(c(crds1[1:2,1], board1[1:3,1]), c(crds1[1:2,2], board1[1:3,2])) ## what you have
x1 = handeval(c(board1[1:3,1]),c(board1[1:3,2])) ## what's on the board
y = straightdraw1(c(crds1[1:2,1], board1[1:3,1]))

13. z = flushdraw1(c(crds1[1:2,2], board1[1:3,2]))
topcard1 = max(board1[1:3,1])
a7 = runif(1) ## random number uniformly distributed between 0 and 1
a8 = (1:numattable1)[roundbets[,1] == roundbets[ind1,1]] ## others who can still bet with you
## The next 5 lines may seem weird, but the purpose is explained in the next comment:
a9 = a8 - dealer1
for(i in 1:length(a9)) if(a9[i]<.5) a9[i] = a9[i] + numattable1
a10 = ind1 - dealer1
if(a10 < .5) a10 = a10 + numattable1
a11 = 2*(a10 == max(a9)) ## So a11 = 2 if you're last to decide; otherwise a11 = 0.
if((x1 > ) && (x < )){
if(a2 < a6) a1 = a2
} else if((x1 > ) && (x < )){
} else if(x > ^3*topcard1){
a1 = max(a5,a2)
} else if((a2 < a6) && ((a7 < .20) || ((a7 < .50) && (a11>1)))){
a1 = a6 }
if((y == 4) || (z == 4)) a1 = max(a6, a2)
if(x > ) a1 = a2

14. if(round1 == 3){ ## after turn:
## If there's a pair on the board and you don't have a set, then check/call up to small bet.
## Same thing if there's 3-of-a-kind on the board and you don't have a full house or more.
## Otherwise, if you have top pair or better, go all in.
## If you had top pair or overpair but now don't, then check/call a medium bet but fold to more.
## If you have an inside straight draw or flush draw then check/call a medium bet as well.
## Otherwise check/fold.
a6 = min(1/3*sum(roundbets[,1:2]),mychips1) ## small bet (1/3 of prev round's pot size)
a5 = min(.75*sum(roundbets[,1:2]),mychips1) ## medium bet (3/4 of prev round's pot)
x = handeval(c(crds1[1:2,1], board1[1:4,1]), c(crds1[1:2,2], board1[1:4,2])) ## what you have
x1 = handeval(c(board1[1:4,1]),c(board1[1:4,2])) ## what's on the board
y = straightdraw1(c(crds1[1:2,1], board1[1:4,1]))
z = flushdraw1(c(crds1[1:2,2], board1[1:4,2]))
topcard1 = max(board1[1:4,1])
oldtopcard1 = max(board1[1:3,1])
if((x1 > ) && (x < )){
if(a2 < a6) a1 = a2
} else if((x1 > ) && (x < )){
} else if(x > ^3*topcard1){
a1 = mychips1} else if(x > ^3*oldtopcard1){
if(a2 < a5) a1 = a2
} else if((y == 4) || (z == 4)){ if(a2 < a5) a1 = a2 } }

15. if(round1 == 4){ ## after river:
## If there's a pair on the board and you don't have a set, then check/call up to small bet.
## Same thing if there's 3-of-a-kind on the board and you don't have a full house or more.
## Otherwise, if you have two pairs or better, go all in.
## If you have one pair, then check/call a small bet.
## With nothing, go all-in with probability 10%; otherwise check/fold.
a6 = .45+runif(1)/10 ## random number between .45 and .55
a5 = min(a6*sum(roundbets[,1:3]),mychips1) ## small bet~ 1/2 of pot size; varies randomly
x = handeval(c(crds1[1:2,1], board1[1:5,1]), c(crds1[1:2,2], board1[1:5,2]))
x1 = handeval(c(board1[1:5,1]),c(board1[1:5,2])) ## what's on the board
if((x1 > ) && (x < )){
if(a2 < a5) a1 = a2
} else if((x1 > ) && (x < )){
} else if(x > ){
a1 = mychips1
} else if(x > ){
} else if(runif(1)<.10){
a1 = mychips1} }
round(a1)
} ## end of zelda