Transcript GOLO: Analyzing the Probabilities of a Dice-Based

Using the On-line Dice-Based Golf
Game GOLO to Illustrate Probability
by
John Gabrosek &
Paul Stephenson
Grand Valley State University
Department of Statistics
March 23, 2010
Rules of Golf



A round of golf consists of 18 holes split
into the front nine and the back nine.
The object on any hole is to get the ball
from the tee to the green and into the hole
in as few strokes as possible.
Golf holes are either par 3, par 4, or par 5.
The par is the expected number of strokes
that it will take for a very good golfer to hit
the ball from the tee into the hole.
Golf Terminology – Good Stuff


Eagle – getting the ball into the hole in two
less strokes than par
In GOLO an eagle is denoted with a star.
Golf Terminology – Good Stuff


Birdie – getting the ball into the hole in one
less stroke than par
In GOLO a birdie is denoted with a circle.
Golf Terminology – Good Stuff


Par – getting the ball into the hole in the
expected number of strokes
In GOLO a par is denoted with a square.
Golf Terminology – Bad Stuff



Bogey - getting the ball into the hole in
one more stroke than par.
Double Bogey - getting the ball into the
hole in two more strokes than par.
Triple Bogey - getting the ball into the
hole in three more strokes than par.
ETC………..
Playing GOLO


Each “golfer” is given nine 12-sided dice to
roll. There are two par 3 dice, five par 4
dice, and two par 5 dice.
The dice are placed into the cup, shaken
up, and rolled onto a flat surface.
Playing GOLO



Each time the dice are rolled the golfer
MUST remove at least 1 die. Any dice
removed are then set aside and not used
in subsequent rolls.
A player continues to roll until he or she
has used all nine dice and thus has
completed nine holes.
A player’s nine-hole score is the sum of
the nine dice that have been set aside.
GOLO Demonstration
Playing the game of GOLO
Classroom Application related to
Strategies & Descriptive Statistics

One potential activity that can be developed allows
students to compare the following two strategies:


Strategy 0 (No strategy): Each student rolls all nine GOLO
dice and sums them to obtain a 9-hole score. Repeat this
twice to obtain an 18-hole score.
Strategy 2 (Lowest die): Each student rolls all nine GOLO
dice and removes the best die relative to par. (If there are
multiple dice with the same score relative to par, then any
one, but only one, of the dice is removed.) The remaining
dice are placed in the cup and rolled again. This procedure
continues until all nine dice have been removed. The scores
for two nines are summed to obtain an 18-hole score.
Classroom Application

Two example games with strategies 0 & 2:
Strategy 0
(No-Strategy)
Strategy 2
(Lowest Die)
Game
Front
9
Back
9
Score
Front
9
Back
9
Score
1
50
57
107
32
33
65
2
46
44
90
40
35
75
Classroom Application

Stem-and-leaf display for simulated results
(60 trials with each strategy)
Strategy 2
Strategy 0
6
124
6
5555667777789999
7
000000001111111122233334
7
5556666777788999
2
8
3
996
8
4444333322211100
9
999987766666555
9
444333332211000
10
98776655
10
00
11
Classroom Application

This application can be used to introduce a variety
of descriptive and inferential procedures for two
independent samples including:



side-by-side boxplots
confidence intervals for independent samples
hypothesis testing for independent samples
The GOLO Dice

There are two red par 3 dice. We call
these die 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
Par 3B – 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7

The eagle/birdie are denoted with a
yellow box, and the pars are denoted with
a green box.
The GOLO Dice

There are five identical white par 4 dice.
The twelve equally-likely faces on the dice
are numbered as follows:
Par 4 – 3, 4, 4, 4, 5, 5, 6, 6, 7, 7, 7, 8
The GOLO Dice

There are two blue par 5 dice. We call
these die 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
Par 5B – 4, 5, 5, 5, 6, 6, 7, 7, 8, 8, 8, 9
Binomial distribution


Consider the fact that each roll can be
thought of as a series of trials where a
success on any given die (or trial) can be
defined as getting a par or better on the
up-face.
Suppose that N dice are rolled, the
probability that X dice are par or better
follows a binomial distribution.
Geometric distribution


Consider the fact that many games end by
needing to roll exactly one die. Suppose
that your competitor inexplicably leaves the
table when you have one die left to roll and
you are in need of at least a par to win.
Let R denote the number of rolls it will take
for you to roll a par or better. Then R
follows a geometric distribution.
Discussion regarding GOLO
Any questions ???