CS173: Intro to Computer Science - Estimating Pi with Monte Carlo Simulation (100 Points)

Assignment Goals

The goals of this assignment are:

1. To use iteration to solve problems while avoiding the need to write duplicate code

Background Reading and References

Please refer to the following readings and examples offering templates to help get you started:

• Iteration Activity

The Assignment

In this lab, you will use iteration to help you avoid writing (and debugging, and fixing, and maintaining) the same code over and over again. Specifically, we will estimate the value of \(\pi\) using iteratively generated random numbers.

Imagine a unit square with a circle inscribed. The diameter of the circle is 1, since the length of a side of the unit square (in which the circle is inscribed) is 1.

Question: what is the area of the square, and what is the area of the circle?

See this article for an animation showing the estimation of the value of \(\pi\) as the ratio of the number of points that fall inside of the area of the circle to the number of points that fall inside the area of the square.

What to Do

Generate 10 pairs of random numbers (x and y coordinates) between 0 and 1. Only some (or perhaps none!) of these points lie inside the circle, but all of these points lie inside the unit square. The ratio of the points inside the circle to the points inside the square should approximate the ratio of the area of the circle to the area of the square, given enough points.

\(\frac{A_{circle}}{A_{square}} = \frac{\pi \times r^{2}}{s^{2}}\)

Since we’re using a unit square with side length \(s = 1\), we know that \(r = 0.5\). This gives us the following:

\(\frac{A_{circle}}{A_{square}} = \frac{\pi \times 0.5^{2}}{1^{2}}\)
\(\frac{A_{circle}}{A_{square}} = \pi \times 0.25\)

Amazingly, the ratio of data points inside the circle to those inside the square is \(\frac{\pi}{4}\). In fact, since every data point is inside the square (because x and y are both between 0 and 1), the percentage of data points inside the unit circle will approximate \(\frac{\pi}{4}\).

We know that a point (x, y) lies within a unit circle if \(x^{2} + y^{2} \leq 1\). Thus, you can create a program with a function estimatePi that accepts a parameter that specifies the number of pairs of random numbers (x, y) to generate. Return the percentage of those random pairs that lie within the unit circle.

Call this function for various numbers of iterations (for example, 10, 100, 1000, 10000, 100000, 1000000, and 10000000), and compute the error from the actual value of \(\frac{\pi}{4}\). Now, run it a second time: do the error values change? If so, by how much?

Here is an example loop that you might include in your estimatePi function, to get you started. I recommend accepting an int parameter in estimatePi called iterations that indicates how many times to run the loop each time, as follows:

public static double estimatePi(int iterations) {
    int inCircle = 0;
    
    for(int i = 0; i < iterations; i++) {
        // TODO: Compute a random x value using Math.random()
        
        // TODO: Compute a random y value using Math.random()
        
        // TODO: Compute x-squared and y-squared
        
        // TODO: if x-squared added to y-squared is less than or equal to 1, increment a counter inCircle 
        // ... (in other words, set inCircle equal to inCircle plus 1)
        
    }
    
    // TODO: your estimate of pi is equal to four times the counter divided by iterations
    // ... but watch out for integer division!  Consider multiplying one of the integers by 1.0 
    // ... during the division to ensure that you use floating point values that do not round!
    // ... if you get 3.0 as your estimate of pi, you might be integer dividing here by mistake.
    
    // TODO: Return the estimate of pi
}

public static void main(String[] args) {
    double piEstimate = estimatePi(1000); // use 1000 iterations this time
    
    // TODO: Your error is Math.PI - piEstimate: print this error, your number of iterations, 
    // ... and your estimate of pi
    
    // TODO: Try putting this main code into a loop of its own, that starts with 10 iterations, and 
    // multiplies by 10 each time through the loop, until you have performed 10000000 iterations.  
    // This way, you will automatically run all your trials for you!
    // ... Try using a for loop that counts from i = 10 up to i = 1000000, with i*=10 to multiply i by 10
    // ... all your code inside main (the call to piEstimate, and your print statements) will go inside 
    // ... this for loop!
}

Finishing Touches and Writeup

Don’t forget to test your program with several different inputs to help verify that things work the way you expect! Think in terms of trying to break your program; if it’s really hard to “trick” your program into getting the wrong answer, you’ve probably done a good job making your code robust.

Also, check the Style Guide to make sure that you’ve written high quality code; make sure your code is “readable,” well indented, uses good variable names, and includes good comments throughout the program.

When you’re done, write a README for your project, and save all your files, before exporting your project to ZIP. In your README, answer any bolded questions presented on this page. In addition, write a few paragraphs describing what you did, how you did it, and how to use your program. If your program requires the user to type something in, describe that here. If you wrote functions to help solve your problem, what are they, and what do they do? Imagine that you are giving your program to another student in the class, and you want to explain to them how to use it. What would you tell them? Imagine also that another student had given you the functions that you wrote for your program: what would you have wished that you knew about how to call those functions?

Exporting your Project for Submission

Here is a video tutorial describing how to write a README for your project, and how to export it. Be sure to save your README file before exporting the project, so that your work is included in the submission!

Submission

In your submission, please include answers to any questions asked on the assignment page in your README file. If you wrote code as part of this assignment, please describe your design, approach, and implementation in your README file as well. Finally, include answers to the following questions:

• Describe what you did, how you did it, what challenges you encountered, and how you solved them.
• Please answer any questions found throughout the narrative of this assignment.
• If collaboration with a buddy was permitted, did you work with a buddy on this assignment? If so, who? If not, do you certify that this submission represents your own original work?
• Please identify any and all portions of your submission that were not originally written by you (for example, code originally written by your buddy, or anything taken or adapted from a non-classroom resource). It is always OK to use your textbook and instructor notes; however, you are certifying that any portions not designated as coming from an outside person or source are your own original work.
• Approximately how many hours it took you to finish this assignment (I will not judge you for this at all...I am simply using it to gauge if the assignments are too easy or hard)?
• Your overall impression of the assignment. Did you love it, hate it, or were you neutral? One word answers are fine, but if you have any suggestions for the future let me know.
• Any other concerns that you have. For instance, if you have a bug that you were unable to solve but you made progress, write that here. The more you articulate the problem the more partial credit you will receive (it is fine to leave this blank).

Assignment Rubric

Description	Pre-Emerging (< 50%)	Beginning (50%)	Progressing (85%)	Proficient (100%)
Algorithm Implementation (60%)	The algorithm fails on the test inputs due to major issues, or the program fails to compile and/or run	The algorithm fails on the test inputs due to one or more minor issues	The algorithm is implemented to solve the problem correctly according to given test inputs, but would fail if executed in a general case due to a minor issue or omission in the algorithm design or implementation	A reasonable algorithm is implemented to solve the problem which correctly solves the problem according to the given test inputs, and would be reasonably expected to solve the problem in the general case
Code Indentation and Spacing (10%)	Code indentation and spacing are generally inappropriate or inconsistent	Code indentation or spacing are generally appropriate but inconsistent in a few isolated instances	Code indentation or spacing are appropriate or consistent, with minor adjustments needed	Code indentation and spacing are appropriate and consistent
Code Quality (10%)	Prior code quality feedback and style guide standards are not reflected in the submitted code to a great extent	Code quality conforms to several standards in the course Style Guide, and progress is demonstrated in improving code quality from prior feedback	Code quality conforms to the standards in the course Style Guide to a great extent, with a few identified areas of improvement	Code quality substantially conforms to the standards in the course Style Guide
Code Documentation (10%)	Code commenting and structure are absent, or code structure departs significantly from best practice	Code commenting and structure is limited in ways that reduce the readability of the program; specifically, javadoc style comments are present for some functions	Code documentation is present that re-states the explicit code definitions	Code is documented at non-trivial points in a manner that enhances the readability of the program; specifically, javadoc style comments are present for all functions
Writeup and Submission (10%)	An incomplete submission is provided, or the README file submitted is blank	The program is submitted, but not according to the directions in one or more ways (for example, because it is lacking a readme writeup or missing answers to written questions)	The program is submitted according to the directions with a minor omission or correction needed, including a readme writeup describing the solution and answering nearly all questions posed in the instructions	The program is submitted according to the directions, including a readme writeup describing the solution and answering all questions posed in the instructions

Please refer to the Style Guide for code quality examples and guidelines.