1. Standard undergrad probability course, not a poker strategy guide nor an endorsement of gambling. The usual undergrad topics + random walks, luck and skill, simulation, computer-based computation, and a few other specialized topics.
Instead of balls and urns, the examples involve Texas Hold’em.
Real examples.

2. 1. Does poker belong in the University curriculum? Is it too frivolous?
Poker is a blend of strategy and chance, and examples can be explored without too much background information.
Poker is an intellectual enterprise, and there is a rich history of card games and gambling problems having been used to motivate many important topics.
The Law of Large Numbers in Bernoulli's Ars Conjectandi (1713) used card games as the main example.
de Finetti's Theorem in La Prévision: ses lois logiques, ses sources subjectives (1937) used gambling and betting odds as the running example.
Why use just one example?
This decreases the time spent on background information.
It takes some work to find real examples in a variety of applications.

3. 2. Samples of conventional probability topics.
High Stakes Poker, Negreanu and Hansen.
Which is more likely, given no info about your cards:
flopping 3 of a kind, or eventually making 4 of a kind?
Consider all combinations of 5 cards, including your 2 cards and the flop.
There are C(52,5) different combinations, all equally likely.
P(flop 3 of a kind) = # of different 3 of a kinds / C(52,5)
How many different 3 of a kinds are possible?
There are 13 x C(4,3) different choices for the triplet.
For each such choice, there are C(12,2) choices left for the numbers on the other 2 cards, and for each of these numbers, there are 4 possibilities for its suit.
So, P(flop 3 of a kind) = 13 x C(4,3) x C(12,2) x 4 x 4 / C(52,5)
~ 2.11%, or 1 in 47.3.
P(flop 3 of a kind or a full house) = 13 x C(4,3) x C(48,2) / C(52,5)
~ 2.26%, or 1 in 44.3.
P(eventually make 4-of-a-kind)?
Again, just forget card order, and consider all collections of 7 cards.
Among C(52,7) equally likely combinations, how many of them involve 4-of-a-kind?
13 choices for the 4-of-a-kind.
For each such choice, there are C(48,3) possibilities for the other 3 cards.
So, P(4-of-a-kind) = 13 x C(48,3) / C(52,7) ~ %, or 1 in 595.

4. Pareto random variables, ch6.6
Pareto random variables are an example of heavy-tailed random variables, meaning they have very large outliers much more frequently than other distributions like the normal or exponential.
For a Pareto random variable, the pdf is f(y) = (b/a) (a/y)b+1, and the cdf is
F(y) = 1 - (a/y)b,
for y>a, where a>0 is the lower truncation point, and b>0 is a parameter called the fractal dimension.

5. The range +/- 1. 96 (s/√n) is a 95% confidence interval for µ. 1
The range +/ (s/√n) is a 95% confidence interval for µ (s/√n)
(from fulltiltpoker.com:)
Based on the data, can we conclude Dwan is a better player? Is his longterm avg. µ > 0?
Over these 39,000 hands, Dwan profited $2 million. $51/hand. sd ~ $10,000.
95% CI for µ is $51 +/ ($10,000 / √39,000) = $51 +/- $99 = (-$48, $150).
Results are inconclusive, even after 39,000 hands!
4. Sample size calculation. How many more hands are needed?
If Dwan keeps winning $51/hand, then we want n so that the margin of error = $51.
1.96 (s/√n) = $51 means 1.96 ($10,000) / √n = $51, so n = [(1.96)($10,000)/($51)]2 ~ 148,000, so about 109,000 more hands.

6. 3. A few unconventional probability topics. Luck and skill in poker.
Define equity gain as expected profit, assuming no future betting.
One may define luck as the equity gained during the dealing of the cards.
Skill = equity gained during the betting rounds.
Hand
Minieri's
cards
Lederer's
Betting actions
Minieri's luck gain
Minieri's skill gain 1
6♠ 6♦
A♣ 7♠
L raises from 1600 to 4300, M raises to 47800, L folds.
206.88 12
7♦ 3♥
A♦ 4♦
L raises to 4300, M raises to 11500, L folds. 16
A♣ J♦
5♠ 5♥
L calls 1000, M raises all in for 26,800, L calls. The board is
3♠ 9♠ K♠ 10♦ 9♦. 18
10♠ 6♥
5♠ 5♣
L calls 1000, M checks.
Flop 7♣ 8♣ Q♥. M checks, L bets 2000, M calls. Turn J♥.
M bets 4000, L folds. 20
7♣ 2♠
Q♠ 9♠
M raises to 6000, L calls Flop A♦ A♠ Q♦. L checks, M bets 6000, L calls. Turn J♣. L checks, M bets 14,000, L raises all in for 35,800, M folds. 21
10♥ 3♦
Q♥ J♠
M calls 1000, L checks. Flop 8♠ 4♥ J♣ . L checks, M bets 2000, L raises to 7500, M raises to 18,500, L raises all-in, M folds. 27
A♣ 5♠
Q♣ 9♣
L goes all in for 29,200, M calls. Board is 7♣ 6♣ 10♠ Q♠ 6♦.
Total PV
52.73%
47.27%

7. Random Walks.
Suppose you start with 1 chip at time 0 and that your tournament is like a simple
random walk, but if you hit 0 you are done. P(you have not hit zero by time 47)?
Starting at 0, P(Y1 ≠ 0, Y2 ≠ 0, …, Y2n ≠ 0) = P(Y2n = 0).
So, for a random walk starting at (0,0), by symmetry,
P(Y1 > 0, Y2 > 0, …, Y48 > 0)
= ½ P(Y1 ≠ 0, Y2 ≠ 0, …, Y2n ≠ 0)
= ½ P(Y48 = 0) = ½ C(48,24)(½)48.
Also P(Y1 > 0, Y2 > 0, …, Y48 > 0) = P(Y1 = 1, Y2 > 0, …, Y48 > 0)
= P(start at 0 and win your first hand, and then stay above 0 for at least 47 more hands)
= P(start at 0 and win your first hand) x P(from (1,1), stay above 0 for ≥ 47 more hands)
= 1/2 P(starting with 1 chip, stay above 0 for at least 47 more hands).
So, multiplying both sides by 2,
P(starting with 1 chip, stay above 0 for at least 47 hands) = C(48,24)(½)48 = 11.46%.

8. Probability calculation using R.
Example 8.3. On one interesting hand from Season 2 of High Stakes Poker, Corey Zeidman (9 9) called $800, Doyle Brunson (Q 10) raised to $6200, and Eli Elezra (10 10), Daniel Negreanu (K J), and Zeidman all called. Use R to calculate the probability that after the 3 flop cards are dealt, Elezra would be ahead, in terms of the best current 5-card poker hand, of his 3 opponents.
Answer. With 8 cards belonging to the 4 players removed from the deck, there are
C(44,3) = 13,244 possible flop combinations, each equally likely to occur. One can use R to imitate dealing each of these flops, and seeing if Elezra is leading on the flop for each. After loading the functions in holdem, one may use the code below to solve this problem.
n = choose(44,3); result = rep(0,n); a1 = c(8,22,34,35,48,49,50,51)
a2 = c(1:52)[-a1]; i= 0
for(i1 in 1:42){for(i2 in ((i1+1):43)){for(i3 in ((i2+1):44)){
flop1 = c(a2[i1],a2[i2],a2[i3]); flop2 = switch2(flop1)
b1 = handeval(c(10,10,flop2$num),c(3,2,flop2$st))
b2 = handeval(c(9,9,flop2$num),c(3,1,flop2$st))
b3 = handeval(c(12,10,flop2$num),c(4,4,flop2$st))
b4 = handeval(c(13,11,flop2$num),c(4,4,flop2$st))
i = i+1; if(b1 > max(b2,b3,b4)) result[i] = 1}}
cat(i1)}
sum(result > 0.5)
This code loops through all 13,244 possible flops and finds that Elezra is ahead on 5,785 out of 13,244 flops. Thus the probability is 5,785/13,244 ~ 43.68%.
Incidentally, in the actual hand, the flop was
6 9 4, Zeidman went all in for $41,700, Elezra called, and Zeidman won after the turn and river were the uneventful 2 and 2.

9. Estimating probabilities using simulations.
Example 8.4. On High Stakes Poker, Daniel Negreanu has been on the losing side of some unfortunate situations where both he and his opponent had extremely powerful hands. For instance, in one hand against Gus Hansen, Negreanu had 6 6, Hansen had 5 5, and the board came 9 6 5 5 8. On another hand, against Erick Lindgren, Negreanu had 10 9, Lindgren had 8 8, and the board came Q 8 J 8 A. Such hands are sometimes called coolers. If, for simplicity, we assume that two players are both all-in heads up before their cards are dealt and we define a cooler as any hand where the two players both have straights or better, then what is the probability of a cooler? Perform 10,000 simulations in R to approximate the answer.
Answer. After loading the holdem package, the code below may be used to approximate a solution to this problem.
One run of this code resulted in 2,505 of the 100,000 simulated hands being coolers. (Different runs of the same code will yield slightly different results.) Thus, we estimate the probability as 2,505/100,000 = 2.505%. Using the central limit theorem, a 95% confidence interval for the true probability of a cooler is thus
2.505% ± √(2.505% % ÷ 100,000)
~ 2.505% ± 0.097%.
n = 10000; a = rep(0,n)
for(i in 1:n){
x1 = deal1(2)
b1 = handeval(c(x1$plnum1[1,],x1$brdnum1), c(x1$plsuit1[1,],x1$brdsuit1))
b2 = handeval(c(x1$plnum1[2,],x1$brdnum1), c(x1$plsuit1[2,],x1$brdsuit1))
if(min(b1,b2) > ) a[i] = 1
if(i/1000 == floor(i/1000)) cat(i) }
sum(a>.5)

10. Competition.
name1 = c("gravity","tommy","ursula","timemachine","vera","william","xena")
decision1 = list(gravity, tommy, ursula, timemachine, vera, william, xena)
tourn1(name1, decision1, myfast1 = 2)

11. 4. Conclusions.
* All the standard probability topics can be motivated using poker.
* Some unconventional topics, like luck and skill, random walks, computation of combinatoric problems, and simulation, also fit in nicely.
* Using real examples is important.