How LO can you GO? Using the Dice-Based Golf Game GOLO to Illustrate Inferences on Proportions and Discrete Probability Distributions

Paul Stephenson
Mary Richardson
John Gabrosek
Diann Reischman
Grand Valley State University

Journal of Statistics Education Volume 17, Number 2 (2009), www.amstat.org/publications/jse/v17n2/stephenson.html

Copyright © 2009 by Paul Stephenson, Mary Richardson, John Gabrosek, and Diann Reischman all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the authors and advance notification of the editor.

Key Words: Active learning; Statistics in sports; Binomial distribution; Sampling distribution of a sample proportion; Confidence interval for a proportion; Hypothesis test on a proportion; Geometric distribution; Hypergeometric distribution; Negative binomial distribution; Distribution of the first order statistic; Theoretical and empirical probabilities.

This paper describes an interactive activity that revolves around the dice-based golf game GOLO.

The GOLO game can be purchased at various retail locations or online at igolo.com. In addition, the game may be played online free of charge at igolo.com. The activity is completed in four parts. The four parts can be used in a sequence or they can be used individually. Part 1 illustrates the binomial distribution. Part 2 illustrates the sampling distribution of the sample proportion. Part 3 illustrates confidence intervals for a population proportion. Part 4 illustrates hypothesis tests for a population proportion.

Extensions of the activity can be used to illustrate discrete probability distributions (including the geometric, hypergeometric, and negative binomial) and the distribution of the first order statistic. The activity can be used in an AP statistics course or an introductory undergraduate statistics course. The extensions of the activity can be used in an intermediate undergraduate statistics course or a mathematical statistics course. Each extension is self-contained and can be carried out without having worked through other extensions or any of the four parts of the main activity.

1. Introduction

In this paper, we discuss an interactive activity that we use to illustrate inferences for proportions, a variety of discrete probability distributions, and the distribution of the first order statistic. The activity is based upon playing the dice-based golf game GOLO. The GOLO game can be purchased online at igolo.com or at various retail locations. Additionally, one very appealing aspect of the use of the GOLO game is that the game may be played online free of charge at igolo.com.

We first provide some general background on the game of golf and the rules of play for the GOLO game.

1.1 Background on Golf

The Merriam-Webster online dictionary defines golf as "A game in which a player using special clubs attempts to sink a ball with as few strokes as possible into each of the 9 or 18 successive holes on a course." Golf is played on a tract of land designated as the course. Players walk (or often drive in motorized electric carts) over the course, which consists of a series of holes. A hole means both the hole in the ground into which the ball is played (also called the cup), as well as the total distance from the tee (a pre-determined area from where a ball is first hit) to the green (the area surrounding the actual hole in the ground). Most golf courses consist of 9 or 18 holes.

The first stroke on each hole is made or hit from the tee, where the grass is well tended to facilitate the tee shot. After teeing off, a player hits the ball again from the position at which it came to rest, either from the fairway (where the grass is cut so low that most balls can be easily played) or from the rough (where the grass is cut much longer than fairway grass, or which may be uncut) until the ball is hit into the cup. Many holes include hazards, which may be of two types:  water hazards (lakes, rivers, etc. ) and bunkers (sand). Special rules apply to playing balls that come to rest in a hazard that make it undesirable to hit a ball into one of the hazards.

At some point on every hole, each player hits her ball onto the putting green. The grass of the putting green (or more commonly the green) is cut very short so that a ball can roll easily. The cup, which is always found within the green, has a diameter of 4.25 in. and a depth of 3.94 in. The cup usually has a flag on a pole positioned in it so that it may be seen from some distance, but not necessarily from the tee. This flag and pole combination is often called the pin. Once on the green, a player putts the ball into the cup in as few strokes as possible.

A hole is classified by its par. Par is the maximum number of strokes that a skilled golfer should require to complete the hole. A skilled golfer expects to reach the green in two strokes less than par and then use two putts to get the ball into the hole. For example, a skilled golfer expects to reach the green on a par four hole in two strokes, one from the tee, another to the green, and then roll the ball into the hole with two putts. Traditionally, a golf hole is either a par three, four, or five. The par of a hole is primarily, but not exclusively, determined by the distance from tee to green. A typical length for a par three hole is anywhere between 100 to 250 yards. A par four is generally between 251 to 475 yards. Par five holes are typically at least 476 yards, and can be as long as 600 yards. Many 18-hole courses have approximately four par-three, ten par-four, and four par-five holes. As a result, the total par of an 18-hole course is usually around 72. One’s score on a hole relative to par is given a nickname. Figure 1 displays the nicknames for the common scoring outcomes.

Figure 1. Some Common Golf Scores

Score relative to par

Nickname

Definition

-2

Eagle

two strokes under par

-1

Birdie

one stroke under par

0

Par or Even

strokes equal to par

+1

Bogey

one stroke over par

+2

Double bogey

two strokes over par

+3

Triple bogey

three strokes over par

1.2 Background on GOLO

1.2.1. History of GOLO

It all started at an Irish pub in Los Gatos, California. Patrick Shea, a local PGA professional, was playing standard dice games with his buddies. He was intrigued with the possibility of playing golf with dice, so he placed 9 standard dice in a cup and within minutes had created the basic rules of the game now called GOLO. The response from friends and family was overwhelming. As the game grew in popularity, a few rules and some new features were added. For further information on GOLO, see: igolo.com.

1.2.2. What is GOLO?

GOLO consists of 9 dice, a dice cup, the rules of GOLO, scorecards, and a pencil. Figure 2 shows an image of the GOLO game.

 

Figure 2. The GOLO Game

The goal of the game, as in real golf, is to shoot the lowest possible score, or "go low!"  The 9 dice represent 9 golf holes on a typical golf course. Each die has twelve sides with various scores on each side - some great, some not so good. Players roll and remove dice to "score."  One can play a variety of games depending on the number of players involved and the length of time available.

There are two par 3 dice (which are red), five par 4 dice (which are white), and two par 5 dice (which are blue).

We call the two par 3 dice Par 3A and Par 3B. The twelve equally-likely faces on the dice are numbered as follows:

Par 3A – 1, 3, 3, 3, 4, 4, 5, 5, 6, 6, 6, 8 and

Par 3B – 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7.

All five par 4 dice are the same. The twelve equally-likely faces on the par 4 dice are numbered as follows:

Par 4 – 3, 4, 4, 4, 5, 5, 6, 6, 7, 7, 7, 8.

We call the two par 5 dice Par 5A and Par 5B. The twelve equally-likely faces on the dice are numbered as follows:

Par 5A – 3, 5, 5, 5, 6, 6, 7, 7, 8, 8, 8, 10 and

Par 5B – 4, 5, 5, 5, 6, 6, 7, 7, 8, 8, 8, 9.

For convenience, on each of the die, a par score is outlined by a square, a birdie is outlined by a circle, and an eagle is outlined by a star.

1.2.3. How to Play GOLO

The basic rules of GOLO are very simple:

1.2.4. The GOLO Web Site

We have obtained numerous GOLO game sets and when we ask students to collect data based upon GOLO we circulate one game set for each student in a class of 30. An alternative to purchasing a large number of GOLO game sets is to ask students to collect GOLO data online. The GOLO web site contains a fully functional interactive version of the GOLO game (upon visiting the web site, igolo.com, the user can select the link: Play the online version of GOLO).

2. The Activity

The activity is completed in four parts. The four parts of the activity can be used in a sequence or they can be used individually. In each of the subsequent sections we will describe the implementation of the four parts of the activity. Each part will require approximately one hour of class time.

For each part of the activity students work individually or in teams of two or three students. Each student or team needs a GOLO game set or access to the Internet so that the GOLO game may be played online.

2.1 Part 1 -- Illustrating the Binomial Distribution

In this part of the activity, students explore the properties and use of the binomial distribution. Prior to completing this part of the activity, students have been introduced to the concept of a discrete random variable. In addition, students have constructed probability distributions in table form.

Each student needs a copy of the Background Sheet (Appendix A.0) and the Part 1 Worksheet (Appendix A.1)

We begin the activity by introducing the GOLO game. We provide a brief background on golf and we discuss typical golf scores. The Background Sheet contains a summary of golf scoring similar to Figure 1 (as well as a summary of the GOLO game dice). After the brief discussion of golf scoring we mention the origin of the GOLO game and we describe how to play the game. We illustrate the three colors of dice; and we note that a par is outlined on each die by a square, a birdie is outlined by a circle, and an eagle is outlined by a star. We mention that we are interested in examining outcomes of rolling the GOLO dice that produce a result of par or better. Thus, when we roll the dice we are looking for squares, circles, and stars.

We discuss the setting for a binomial random variable. We explain that the binomial setting is characterized by the following:


We explain that the number of successes in the n trials, denoted by X, is called a binomial random variable and is said to have a binomial distribution.

We explain that each roll of a GOLO die can be thought of as a single trial in a binomial experiment where success on a given trial is defined as a specified outcome or better on the up-face of the die. For example, suppose that we define a success as par or better on a die. If n dice are rolled, the probability that X dice are par or better follows a binomial distribution. The probability that n rolls result in x successes can be found as:  where x=0,1,2,…,n and p is the probability of success on each roll; 0 ≤ p ≤ 1.

We ask students to answer a series of questions, assuming that a GOLO player is beginning a "new nine."  First, we ask students to explain why the number of pars or better thrown on the roll of all 9 dice can be considered a binomial random variable. We consider each of the nine dice to represent a separate, independent trial. By examining the GOLO dice we can see that the proportion of rolls that result in an outcome of par or better is 1/3, since on each die, four of the twelve possible equally likely outcomes is par or better than par. Thus, the probability of a success (par or better) on each trial is the same  The random variable of interest is the number of successes X in the n=9 trials.

We ask students to calculate the following probabilities: