Computer Assignment 1

Computer Assignment 1
Deliverable: A Python computer program that uses the simulation approach outlined below as the solution
to the given problem. To earn credit for the assignment the program must satisfy the rubric below.
Topic
Using simulation to solve a probability problem.
Introduction
There may be multiple approaches to solving a math problem. Historically abstract symbolic arguments
were the accepted way of addressing math problems. As computational aids have grown numerical
approaches have become more common; though, purist may view them unfavorably. With the modern
computers numerical approaches have become ubiquitous. When we focus our interest on that branch of
mathematics called probability here too computers can be used to solve problems. The application of
computers to solving a probability problem is sometimes referred to as simulation. The idea is to code the
probability experiment intrinsic to the problem. If this code satisfactorily models the probability
experiment, then it may be repeated for as many trials of the experiment as desired. This provides a way
of simulating the probability experiment for the problem being addressed. Thus if feasible, a
computational solution yielding an answer that approximates the true answer may be obtained.
Example ‘Balls in Cans’ Problem
Three balls are tossed randomly into 5 cans. What is the probability that they all land in different
cans? (Note each of the cans are able to hold from 0 to 3 balls.) Probability and Stochastic
Processes by F. Solomon (1987), page 190.
The problem can be solved using different approaches. The problem is first solved in the traditional
abstract way. In this context there are at least two ways to solve this problem.
First Solution
One way to respond to the problem is with classical probability: the ratio of the number of outcomes in
the event to the number of outcomes in the sample space. We ask ourselves, what are the number of
outcomes in the sample space? Because each ball has five choices and these choices are independent
(lexicographical definition of independent not the probabilistic definition of independent) the answer is 5 ∙
5 ∙ 5 = 125. Similarly, we ask ourselves, what are the number of outcomes in the event? For the event
we want them all in separate cans; so, while one ball has five choices, the other two will have only four or
three choices respectively and the answer is 5 ∙ 4 ∙ 3 = 60. The ratio is the answer; thus, 60⁄125 = 0.48.
Second Solution
Another way to respond to the problems is with conditional probability: The product of the three
conditional probabilities. Then we need to both ask ourselves three questions and answer those questions.
First question, what is the conditional probability of the first ball? Answer, choose any of the five
available cans from the five cans or 5⁄5.
Second question, what is the conditional probability of the second ball? Answer, choose any of the
remaining four available cans from the five cans or 4⁄5.
Third question, what is the conditional probability of the third ball? Answer, choose any of the remaining
three available cans from the five cans or 3⁄5.
The product is the answer; thus, 1 ∙
4
5
∙
3
5
=
12
25
= 0.48.
A computational solution of the problem.
Consider that each ball has a choice of five cans and that the cans are chosen at random. We can label the
three balls (starting at zero for reasons that will become apparent) as, {0, 1, 2}. We can label the five
cans with the numbers one through five, {1, 2, 3, 4, 5}. In the context of computer programming what is
a possible way of doing the assignment of balls into cans?
As the reader is no doubt familiar computer languages typically have arrays (Python calls these lists).
The storage locations in an array (list) are known as elements and each element in an array is assigned a
unique number known as a subscript. So, we can have three elements in the array with indexes 0, 1, and
2. We can have a random number generator provide randomly the numbers 1, 2, 3, 4, and 5. Any one of
these five numbers can be assigned to each of the three elements. What are some possible instances of
these arrays?
[3, 3, 3] three the same
[2, 3, 2] two the same
[4, 1, 1] two the same
[2, 1, 5] all different
[5, 2, 4] all different
When we have written the code to produce all the possibilities then we have modeled the probability
experiment. The goal is to solve the probability problem by using both a computer and a computer
language to create a simulation of the probability problem. The model of the probability experiment
above is part of this simulation. Additionally, we need to separate from all the results those that satisfy
the requirements of the event: all elements (cans) are different. The experiment would be repeated many
times (as big as needed) and the ratio of the successful results to all the results will approach the correct
answer with increasing accuracy (with more repetitions). It is not reasonable for the answer to be exactly
the true answer. The Python program of this solution is provided with this assignment.
Conclusion
Arguably a way of understanding probability theory is to explore both a traditional ‘pencil and paper’
solution and a modern ‘computational’ solution using a fundamentally different approach from the ‘pencil
and paper’ solution. In this way the reader should be able to look back over what has been done and see
how and why the different approaches successfully address the same problem. In closing to evaluate
whether we have been successful the reader is now invited to solve the problem below. A rubric is
provided as a guide to efficiently prepare and submit the necessary work.
Problem ‘Auditor’
Three firms using the same auditor independently and randomly select a month in which to conduct
their annual audits. What is the probability that all 3 months are different? Elementary Statistics by M.
Triola (1997), Page 152.
Rubric
Name and I.D. #
Name of Assignment
Due Date
Submission Date
Not Satisfactory:
Data requested absent.
Satisfactory:
Has all data requested.
A block comment at beginning
of program summarizing it. Line
comments throughout the
program explaining its
operation.
Not Satisfactory:
Conveying incomplete thoughts
or unnecessarily excessive.
Satisfactory:
A brief two or three sentence
explanation.
Identify lines in program that
model the experiment intrinsic
to the probability problem.
Not Satisfactory:
Experiment not identified or
blocked off.
Satisfactory:
Experiment clearly identified.
List references. Not Satisfactory:
No references listed.
Satisfactory:
References – assignment
handout, internet, students,
and etc. listed
Run the program multiple times
with magnitude shifts in the
number of overall repetitions of
the experiment.
Not Satisfactory:
Not run for a sufficient number
of values to show change in
output.
Satisfactory:
Output shows increasing
accuracy with increasing
repetitions of experiment.
Upload the .py file of the
program to and a PDF of the
program with the output.
Not Satisfactory:
Absent files.
Absent output.
Satisfactory:
The two files and the output are
submitted.