Probability for engineering
IEE 2621 – Probability for Engineers Spring 2017
SEMESTER PROJECT – Teams of 3-4 DUE DATE 04/18/17
Project Description:
Suppose that you’re a team of young adventurers who enjoy making bold moves in life. You also
enjoy the concept of chance and decide that your summer trip around the country will obey the
laws of uniform discrete random variables!! For your summer trip you decided to rent a car to
travel around Ameri`ca’s 48 contiguous states. At the beginning of each day of your trip you will
roll an 8-sided die to determine which state to go to next through the use of the attached US
map. To illustrate how to use this map, consider the following two tables, which show the results
of rolling a die if you are currently in Oklahoma (OK) or currently in North Dakota (ND).
Table 1: Result of Rolling 8-Sided Die when Currently in Oklahoma
Die Roll 1
* 2 3 4 5 6 7 8
†
Destination OK TX NM CO KS MO AR OK
Table 2: Result of Rolling 8-Sided Die when Currently in North Dakota
Die Roll 1
* 2 3 4 5 6 7 8
Destination ND SD MT MN CANADA
Project Part I – Two Week Trip Instructions:
Choose any state (besides Michigan) to begin your two week trip. It is recommended to read
Project Part II before deciding upon your initial state. Starting on Day 0, roll an 8-sided die to
determine your movement. Repeat this process until 14 days (movements) have been
completed. For this two week trip, you are not allowed to leave the US (i.e., any roll that would
results in crossing into Canada or Mexico should be considered as a roll of 1). Repeat this
experiment 100 times (recording your results), where each time you start at the same state.
Below is an example of two experiments.
Day 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Dice Roll N/A 7 8 2 8 6 1 3 5 8 8 2 8 8 4
Location AR MS MS LA LA LA LA AR MO KY VA NC NC NC TN
Day 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Dice Roll N/A 7 2 4 8 7 8 6 1 7 8 6 7 6 4
Location AR MS LA MS MS MS MS MS MS MS MS MS MS MS TN
* A roll of 1 will always result in staying in your current state, except for Tennessee (TN) and Missouri (MO). You
will never stay two days in a row in TN or MO.
† Any roll that is not labelled on the map results in staying in your current state.
Discuss the results of these 100 experiments. Things that should be discussed would
include, but
are not limited to:
1) Did you return to your original state; and if so how often did you return?
a. Within each experiment.
b. Among all 100 experiments.
2) Did you return to your original state; and if so how quickly did you return?
3) How often did you end up at the same state at the end of two weeks?
4) What state was visited most often?
5) What state was visited least often?
6) Did you ever pass through Michigan; and if so how often?
Project Part II – The Return Home Instructions:
Use the same initial state that you chose for Part I. Starting on Day 0, roll an 8-sided die to
determine your movement. Repeat this process until you either 1) Arrive in Michigan or 2) Leave
the US. Repeat this experiment 50 times (recording your results), where each time you start at
the same state. Below is an example of one experiment.
Day 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Dice Roll N/A 6 4 6 1 8 4 6 7 4 5 5 5 3 2
Location AR TN NC NC NC NC NC NC NC TN MO NE SD WY CO
Day 15 16 17
Dice Roll 4 2 7
Location UT AZ VIVA LA MEXICO!!
Discuss the results of these 50 experiments. Things that should be discussed would include, but
are not limited to:
1) How many days did it take to either arrive to Michigan or leave the US?
2) How often did you arrive to Michigan compared to how often you left the US?
3) When you left the US, did you go to Canada more often than Mexico (or vice versa)?
4) Were than any significant difference between these experiments and the previous
experiments (e.g. number of different states visited, number of times visiting a specific
state, etc.)?
Project Part III – Discrete Time Markov Chain Analysis:
Using a Discrete Time Markov Chain (DTMC) analyze the two systems from Project Part I and II.
Compare and contrast the theoretical DTMC results with those that were obtained through your
experiments. Submit Project Parts I-III as a single report.
2 3
2 3
4
5
6
7 2 3
4 5 2 3 4
2
3
4
2
3 4
5
6
2
3 4
5
2
3
4
5
6 2 3
4 5 6
7
2 3 4 5 6
7
2 3 4 5
2 3 4
2 3
4 5 6
7
2 3 4 5 6
7
2 3 4 5
2 3 4 5 6
2 3 4
5
2
3 4
2 3
4 5 6 3 2
4
5 6 7
1
2 3
4 5 6
7
2 3
4 5
3 2 4 5
2 3
4
5
6 2 3
4
2
3
4
5 2 3
4
5 6
2 3 2
3 4
5 2 3
4
5 2
3
4 5 6 2
3 1 2 4 3
5 6 7 2 3
4 2 3
4 5 2 3
4 5 6 7
8
8
2 3 5
2 3 4 5
2 3 4
5 6 7
2
3
4
2
3
4
6
4 3 2 5 6
2
3 4
2
3 4 2
2 3 4
2 3
2 3
4 5
WA
OR
CA
TX
NV
AZ NM
UT
MT ID WY
CO
ND
SD
OK
NE
MN WI
IL IA
MO
LA FL
AL GA MS SC
NC
MI
IN OH
WV VA KY
TN
MD PA
NY
DE
NJ
ME
RI
MA CT
NH
VT
KS
AR
4-8 8 6-8 5-8 6-8
5-8
7-8 5-8 5-8
3-8
5-8 6-8 6-8 6-8
8
5
5 7
6 6
7
2 3 5
2 3 4 5
2 3 4
5 6 7
2
3 4
2
3
4
6
4 3 2 5 6
2
3 4
2
3 4 2
2 3 4 2
3
OH
WV VA
MD PA
NY
DE
NJ
ME
RI
MA
CT
VT NH 7-8 5-8 5-8
3-8
