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In the game of draw poker, 5 cards are dealt from a deck of cards. There are nine different types of poker hands in terms of value, defined as follows:

- No Value. The hand is of none of the following types.
- One Pair. There are four distinct denominations in the hand; one denomination occurs twice and each of the other three occurs once.
- Two Pair. There are three distinct denominations in the hand; two of the denominations occur twice each and the remaining denomination occurs once.
- Three of a Kind. There are three distinct denominations in the hand; one denomination occurs three times and the other two denominations occur once each.
- Straight. The denominations of the cards can be arranged in a perfect sequence but the cards are not all in the same suit. An ace can be considered the smallest denomination or the largest denomination.
- Flush. The cards are all in the same suit, but the denominations cannot be arranged to form a perfect sequence.
- Full House. There are two distinct denominations in the hand; one denomination occurs three times and the other two times.
- Four of a Kind. There are two distinct denominations in the hand; one denomination occurs four times and the other occurs once.
- Straight Flush. The cards are all in the same suit and the denominations can be arranged to form a perfect sequence.

To model poker as a random experiment, we first make some simplifying assumptions (as always). First, we are not concerned with the betting aspect of the game or with discarding and drawing additional cards. We do not allow jokers or other wild cards. Finally, we assume that the deck is well-shuffled. With these assumptions, we have a well-defined random experiment, and the value `V` of the hand (taking values 0 through 8 as defined above) is the *random variable* of interest in the experiment.

We will let `D` denote the deck of cards, thought of as a set. Note that `D` naturally has the structure of a Cartesian product set, specifically the product of the set of *denominations* (ace, two through 10, jack, queen, king) with the set of *suits* (clubs, diamonds, hearts, spades). After all, one describes a typical card as the *queen* of *hearts*.

The order of the cards in the hand does not matter in draw poker, so we will record the outcome of our random experiment as an unordered set of 5 cards. Thus, the *sample space* (the set of all outcomes) consists of all possible poker hands:

`S` = {{`x`_{1}, `x`_{2}, `x`_{3}, `x`_{4}, `x`_{5}}: `x _{i}` in

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`. Our basic modeling assumption (based on the fact that the deck is well-shuffled) is that all poker hands are equally likely. Thus, just like in the birthday problem, we give `S` the *uniform probability distribution*:

`P`(`A`) = #(`A`) / #(`S`) for `A` `S`.

Computing the probabilities of the various values of `V` (these probabilities form the *probability density function* or *probability mass function* of `V`) is a good exercise in combinatorial probability. First, note that the number of different poker hands is simply the number of combinations of size 5 from a population of size 52, and hence

#(`S`) = `C`(52, 5) = 2,598,960.

Thus, to find the desired probabilities, we simply need to count the number of poker hands of each type. In turn, this counting will involve the two fundamental rules of combinatorics: the *multiplication rule* and the *addition rule*.

We will start with the event {`V` = 1}, which consists of all poker hands that have one pair (as defined above). To count this set, we will define an algorithm that "constructs" all poker hands with one pair (in one and only one way), and then count the number of ways of performing each step in the algorithm. By the multiplication principle, the number of paths through the algorithm (and hence the number of poker hands with one pair) is the product of the number of ways of performing the various steps. One possible algorithm is given in following list, where the number of ways of performing each step is shown in square brackets.

- Select the denomination that is repeated [13].
- Select 2 cards of the denomination in step 1 [
`C`(4, 2) = 6] - Select the remaining 3 denominations [
`C`(12, 3) = 220]. - Select 1 card of each of the denominations in step 3 [4
^{3}= 64].

Therefore the cardinality of event {`V` = 1} is 13 · 6 · 220 · 64 = 1,098,240 and hence

`P`(`V` = 1) = 1,098,240 / 2,598,960 = 0.422569.

The remaining types of hands can be counted using similar techniques and are left as exercises.

**1.** Show that
`P`(`V` = 2) = 123,552 / 2,598,960 = 0.047539.

**2.** Show that
`P`(`V` = 3) = 54,912 / 2,598,960 = 0.021129.

**3.** Show that
`P`(`V` = 8) = 40 / 2,598,960 = 0.000015.

**4.** Show that
`P`(`V` = 4) = 10,200 / 2,598,960 = 0.003925.
*Hint*: Use the result of Exercise 3.

**5.** Show that
`P`(`V` = 5) = 5,108 / 2,598,960 = 0.001965.
*Hint*: Use the result of Exercise 3.

**6.** Show that
`P`(`V` = 6) = 3,744 / 2,598,960 = 0.001441.

**7.** Show that
`P`(`V` = 7) = 624 / 2,598,960 = 0.000240.

**8.** 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 discrete probability density function (or probability mass 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.

**9.** Find the probability of getting a hand that is three of a kind or better.

The poker applet simulates the poker experiment. The value `V` of the hand is recorded in the first table on each update. The graph shows the discrete probability density function in blue. When the simulation runs, the relative frequency function of `V` is shown in red. The second table shows the same information as the graph, but in numerical form. (Note that the probabilities and relative frequencies in this table are rounded to four decimal places.) The Stop Frequency list box can be set so that the simulation automatically stops when the hand has a particular value.

**10.**
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 defined above.

**11.**
In the poker experiment, note the shape of the density graph. Note that most 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 probability density function..

**12.**
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.

`V`= 3`V`= 4`V`= 5`V`= 6`V`= 7`V`= 8

**13.**
Let us close this page with one last mathematical exercise. In the movie The Parent Trap (1998), both twins get straight flushes on the same poker deal. Show that the probability of this event is 3.913 × 10^{-10}. If the twins were to play one poker hand every second, they could expect to play about 81 years before this event might reasonably occur.

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