Index

# Discrete Mathematics Project Counting Techniques/Probability Activity

## Title

"What're the odds?" (Jim Arnow)

## Goals

1. Students learn to apply discrete counting techniques and concepts from discrete probability via an inherently interesting application -- the odds of various poker hands.
2. Students work in small groups to generate probability values then present their methods to the other members of the class.

## Abstract

This activity deals with computing probabilities of various poker hands. This topic is easily accessible to most students, since they may have played the game themselves, or are easily taught the game. As a group, the basics of calculating such probabilities are discussed with a specific hand, and then the students are left to apply the techniques to various other hands. This activity directly deals with the topic of discrete probability and counting techniques.

## Problem Statement

Discuss the game of poker with your students. The students should be made familiar with the various hands possible in poker as well as a small amount of some of the playing strategy. Included in this strategy should be a discussion of the fact that in poker it is useful if a player knows which hands are more common than others. The more you know, the easier it is to make decisions about various hands. To determine which hands are more common, we can use counting techniques to determine their relative frequencies.

## Instructor Suggestions

Possible lesson plan:
1. Discuss the rules until the students are familiar with how the game is played. If there are some students who are familiar with the rules of the game, have them present them to the whole class.
2. For simplicity, the situation should be restricted to a game consisting of five cards with no draw.
3. Once the students are comfortable with the rules of the game, discuss the number of different possible hands that can be drawn with five cards. This can be presented as an example of the multiplication principle or as a combination. (The total number of possible hands is C(52,5) or 2,598,960.)
4. Discuss the definition of probability in terms of counting -- the number of possible ways of an outcome occurring divided by the total number of outcomes.
5. Calculate, as a large group, the number of possible ways of drawing a representative hand. A full house (total number -- 3,744) and/or a straight flush (total number -- 40) are good examples, since they include some features of other hands.
6. Divide the class into small groups and give each group the task of calculating the number of possible ways of drawing two or three of the possible hands. If possible, it may be best to assign hands to more than one group so that they may compare answers.
7. When all the groups have made sufficient progress, have a spokesperson for each group present their results to the class. Discuss the results as a group and compare the results with the results published in Hoyle or similar books on card games.
Notes:
• These calculations can be very confusing and even intimidating to students. To make them easier to approach, have the students describe various hands in terms of their distinct characteristics. For instance, to uniquely determine a straight flush, we need only determine the lowest card (ten possible ways), and the suit (four possible ways). The total number of straight flushes is then found by the multiplication principle -- 4 times 10 or 40.
• Students can get very confused by the fact that the cards are dealt sequentially. Remind them that a hand is the same regardless of what order the cards are dealt -- we can always sort the cards in our hands and that doesn't affect what the hand is.
• Some useful terms from poker:
Rank:
The numerical value of the card, Ace, 2, 3, 4, 5, 6, 7, 8, 8, 10, Jack, Queen or King.
Suit:
Hearts (H), Clubs (C), Diamonds (D) or Spades (S)
• How to make the various calculations (other explanations are possible):
Straight flush:
4 ways of choosing the suit and ten choices for low (or high) card. Total -- 40 ways.
Four of a kind:
13 choices for the rank, 48 choices for the other card. Total -- 624 ways.
Full House:
Thirteen choices for the rank of the three of a kind, C(4,3) ways to choose three cards of that rank, 12 ways to choose the rank of the pair, C(4,2) ways to choose two cards of that rank. Total -- 3,744 ways.
Flush:
Four choices for the suit, C(13,4) choices for the four cards minus 40 straight flushes which have already been counted. Total -- 5108 ways.
Straight:
Ten choices for the rank of the lowest card times 4 for the choice for the suit of each card minus 40 for the straight flushes which have already been counted. Total 10,200 ways.
Three of a kind:
13 choices for the rank of the three of a kind, C(4,3) ways to choose three cards of that rank, 48 choices for one of the remaining cards, 44 choices for the other (not 47 since it can't be the same rank as the other non-matching card), divide by two for the possible permutations of the two non-matching cards. Total -- 54,912 ways.
Two Pairs:
C(13,2) ways to choose the ranks of the two pairs, C(4,2) ways to choose which cards of the first rank, C(4,2) ways to choose which cards of the second rank and 44 choices for the remaining card. Total -- 123,553 ways.
One Pair:
13 ways to choose the rank of the pair, C(4,2) ways to choose which cards from that rank, 48 choices for the first non-matching card, 44 choices for the next non-matching card, 40 choices for the last non-matching card, divide by P(3,3) or the possible permutations of the three non-matching cards. Total ways -- 1,098,240.
• If the situation arises, the difference between odds and probability can be discussed. Remember that the probability of an event occuring can be calculated as the number of ways that the event can occur divided by the total number of outcomes. The odds of something occuring can be calculated as the ratio of the number of ways that the event can occur to the number of ways in which the event does not occur.

## Materials

decks of cards, "What're the odds?" handout, handout of Hoyle's count of poker hands, calculators, scratch paper

## Time

Discussion of rules of poker (5 min.), discussion of how to calculate count for one example (10 min.), Small group work (30 min. to 1 hour), presentation of small group work and large group discussion (15 min.)

## Mathematics Concepts

### Discrete Mathematics

Counting techniques, probability, permutations and combinations, odds and probability, independent events

### Related Mathematics

factorial, sequences, sets, common factors (for reducing probability calculations)

## NCTM Standards Addressed

Problem Solving, Communication, Reasoning, Connections, Algebra, Probability

## Colorado Model Content Standards Addressed

Number Sense (1), Algebraic Methods (2), Data Collection and Analysis (3), Problem Solving Techniques (5), Linking Concepts and Procedures (6)

## Curriculum Integration

This activity could be integrated:
1. into an algebra class which has covered the definition of factorial
2. into a probability or statistics class as an exposure to a classic "real world" application of probability

## Further Investigation

The concept of calculating probabilities can be extended to a number of similar games, such as roulette, craps, dungeons and dragons, etc. The students should be able to come up with a variety of possible applications.

Some related questions:

1. What is the probability of getting one or the other of some of these hands, e.g. What is the probability of drawing a flush or four of a kind?
2. What are some important probabilities if I am playing five card poker where I am allowed to draw? (i.e. I can return any number of my cards and replace them from the deck.)
3. Do my probabilities change if I am playing with a number of other people and they are dealt their cards before I am? (A: No) Why not?
4. What would change about this problem if we were to draw seven cards and take our best five from them? How could we approach the problem?
5. What other games have probabilities such as this to be calculated?

## References, Resources and Related Web Links

Crisler, N., Fisher, P., & Froelich, G. (1994). Discrete Mathematics Through Applications. New York: W. H. Freeman and Company.

Frey, Richard L. (1964). The New Complete Hoyle. New York: Doubleday.

Ross, S. (1984). A First Course in Probability. New York: Macmillan Publishing.