Xj for each i
and j.Virtual Laboratories > Games of Chance > 1 [2] 3 4 5 6 7 8
A deck of cards naturally has the structure of a product set and thus can be modeled mathematically by
where the first coordinate represents the denomination (ace, two through 10, jack, queen, king) and where the second coordinate represents the suit (clubs, diamond, hearts, spades).
There are many different poker games, but we will be interested in standard draw poker, which consists of dealing 5 cards at random from the deck D. The order of the cards does not matter in draw poker, so we will record the outcome of our random experiment as the random set (hand)
X = {X1, X2,
X3, X4, X5}
where Xi = (Yi, Zi)
in D for each i and Xi Xj for each i
and j.
Thus, the sample space consists of all possible poker hands:
S = {{x1, x2, x3,
x4, x5}:
xi in D for each i and xi
xj for each i
and j}.
Our basic modeling assumption (and the meaning of the term at random) is that all poker hands are equally likely. Thus, the random variable X is uniformly distributed over the set of possible poker hands S.
P(X in A) = #(A) / #(S)
for A 
S.
In statistical terms, a poker hand is a random sample of size 5 drawn without replacement and without regard to order from the population D. For more on this topic, see the chapter on Finite Sampling Models.
There are nine different types of poker hands in terms of value. We will use the numbers 0 to 8 to denote the value of the hand, where 0 is the type of least value (actually no value) and 8 the type of most value. Thus, the hand value V is a random variable taking values 0 through 8, and is defined as follows:
1. Run the
poker experiment 10 times in single-step mode. For each outcome, note
that the value of the random variable corresponds to the type of hand, as given above.
Computing the density function of V is a good exercise in combinatorial probability.
2. Show that the
number of different poker hands is #(S) = C(52, 5) = 2,598,960.
In the following exercises, you will frequently need to use the multiplication rule of combinatorics to count the number of poker hands of a given type. In each case, try to construct an algorithm for generating the poker hands of the given type, and then count the number of ways of performing each step in the algorithm.
4. Show that P(V
= 1) = 1,098,240 / 2,598,960 = 0.422569.
5. Show that P(V
= 2) = 123,552 / 2,598,960 = 0.047539.
6. Show that P(V
= 3) = 54,912 / 2,598,960 = 0.021129.
7. Show that P(V
= 8) = 40 / 2,598,960 = 0.000015.
8. Show that P(V
= 4) = 10,200 / 2,598,960 = 0.003925. Hint: Use the result of Exercise 7.
9. Show that P(V
= 5) = 5,108 / 2,598,960 = 0.001965. Hint: Use the result of Exercise 7.
10. Show that P(V
= 6) = 3,744 / 2,598,960 = 0.001441.
11. Show that P(V
= 7) = 624 / 2,598,960 = 0.000240.
12. Show that P(V
= 0) = 1,302,540 / 2,598,960 = 0.501177. Hint: Use the addition rule of
probability and the previous exercises.
Note that the density function of V is decreasing; the more valuable the type of hand, the less likely the type of hand is to occur. Note also that no value and one pair account for more than 92% of all poker hands.
13. In the
poker experiment, note the shape of the density graph. Note that some of
the probabilities are so small that they are essentially invisible in the graph. Now run
the poker hand 1000 times, updating every 10 runs. Note the apparent convergence of the
relative frequency function to the density function..
14. In the
poker experiment, set the update frequency to 100 and set the stop
criterion to the value of V given below. Note the number of poker hands required.
15. Find the
probability of getting a hand that is three of a kind or better.
16. In the movie The
Parent Trap (1998), both twins get straight flushes on the same poker deal. Find
the probability of this event.
17. Classify V
in terms of level of measurement:
nominal, ordinal, interval, or ratio. Is the expected value of V meaningful?