It may surprise you to learn that random events such as coin flips, dice rolls, and roulette wheel spins are central to many computer science applications, such as cryptography and database management. While events such as these are random in nature, their behavior over the long run is predictable and so can be used to construct deterministic solutions to complex problems. Problem solutions that use such random events are known as Monte Carlo methods, named after the city famous for its casinos.
In class, we used a Monte Carlo method to estimate word frequencies. By generating letter sequences at random and keeping track of the number of words obtained, we were able to estimate the total number of words. In this lab, you will use a related Monte Carlo method to approximate the value of PI.
The first step in approximating PI will be in estimating the ratio of the area of a circle to the area of the square that circumscribes it (see the picture on the right). This can be accomplished using a Monte Carlo method involving random points.
Think of the square as a dart board with the a circle inscribed in it. If you threw darts randomly at the board so that each dart is equally likely to land at any point, then the distribution of the darts would allow you to estimate the ratio of the areas of the circle and square. For example, if you threw 500 darts and 400 of them landed inside the circle, then the area of the circle could be estimated as 400/500 = 0.80 = 80% of the area of the square.
The Web page MontePI.html can be used to simulate the generation of random points (or dart holes) within a square, and also keep track of the number of points that land inside the inscribed circle. By default, 100 random points will be displayed and tallied each time you click on the button. The number of points can be changed, however, by entering a new number in the box at the top.
EXERCISE 1: Load this Web page and perform 10 different experiments. For each experiment, generate 100 random points and list the number of points that land inside the circle. Be sure to clear the display and reset the counts (by clicking on the appropriate button) between experiments. List your results below.
Due to the random nature of these experiments, there will no doubt be some variability in your results. In order to arrive at a single answer, a natural approach is to average the results obtained via repeated experiments. What is the average of your results from the 10 experiments? You may use the online calculator (Start -> Programs -> Accessories -> Calculator) if you wish.
How consistent were your results from the 10 different experiments? That is, did you get close to the same count each time, or were there fairly large differences? One formal way of quantifying consistency is to consider the maximum relative difference between any one result and the average of the results. First, find the single result that differs most from the average of the results. Then, take the difference between those two and divide by the average. For example, suppose you conducted 4 experiments with results 60, 80, 75, and 85. The average of these 4 results is 75. The maximum difference between 75 and any one of the results is 15 (i.e., 75-60), and so the maximum relative difference is 15/75 = 0.20 = 20%. What is the maximum relative difference using the numbers from your 10 experiments?
Monte Carlo methods rely on the fact that while the outcome of a single random event cannot be predicted, the distribution of repeated random events often can be. Consider a coin flip, for example. While it is impossible to predict whether a single flip will produce heads or tails, it is extremely likely that given a large number of flips, the number of heads and tails will be approximately the same. Since there are only two possible outcomes of a coin flip, each equally likely, we say that the expected distribution of heads and tails is 50/50.
A key phrase in determining the expected distribution of random events is "given a large number...". It shouldn't surprise you that in the short run, statistical anomalies can occur. For example, if you flipped a coin twice, you might get two heads or two tails as opposed to the expected distribution of one of each. When the number of events is small, such anomalies can greatly skew statistics (based on two flips, we might conclude that the expected distribution is 100% heads!). In the long run, however, anomalies will balance out (e.g., streaks of heads will be counter-balanced by streaks of tails) and the expected distribution will be achieved. The more flips you make, the closer you are likely to come to the expected 50/50 distribution.
EXERCISE 2: Perform 10 experiments with 1,000 random points and list the results below. What is the average of the 10 results? Are the numbers you obtained more or less consistent than your experiments involving 100 points? What would you expect? Explain your answers.
EXERCISE 3: Perform 10 experiments with 10,000 random points and list the results below. What is the average of the 10 results? Are the numbers you obtained more or less consistent than your experiments involving 1,000 points? What would you expect? Explain your answers.
EXERCISE 4: In the Web page for generating random points, you can add new points to the square and accumulate counts by repeatedly clicking on the button. Thus, you could generate 1,000 random points by performing 10 different experiments of 100 points each, accumulating the point counts as you conduct each experiment. Is this equivalent to conducting a single experiment with 1,000 points? Explain your reasoning. How could you go about testing your claim?
Once you have estimated the ratio of the areas of the circle and the square, it is a relatively simple task to approximate PI. The formulas for the area of a square (side2) and circle (PI*radius2) are taught in high school geometry. In the special case of a circle inscribed in a square, the radius of that circle is (side/2), making its area PI*(side/2)2 = PI*side2/4. Thus, the ratio of the areas of the inscribed circle and the square is:
(area of inscribed circle/area of square) = (PI*side2/4)/side2 = PI/4Solving for PI, we see that:
PI = 4*(area of inscribed circle/area of square)Suppose that you generated a large number of random points, 75% of which landed inside the circle. Estimating that the area of the inscribed circle is 75% of the area of the square, the above formula would then imply that PI is approximately 4 * 0.75 = 3.0.
EXERCISE 5: Create a Web page named approx.html that performs this final computation on experimental data to approximate the value of PI. The page should prompt the user for the results of a particular experiment, i.e., the number of points that landed inside the circle and the total number of points. It should then use that data to approximate the value of PI and display that value in the page.
For the sake of readability, your page should display the number of points inside the circle and the total number of points, as well as the approximation of PI. For example,
750 out of 1000 points were inside the circle. Based on this data, PI is approximately: 3.0
EXERCISE 6: Using the average result from the experiments involving 100 points (EXERCISE 1), use your Web page to approximate PI. How accurate is the result compared to the actual value of PI (3.141592653...)?
EXERCISE 7: Using the average result from the experiments involving 1,000 points (EXERCISE 2), use your Web page to approximate PI. How accurate is the result? Is the approximation more or less accurate than the one made with only 100 points? What would you expect? Explain your answers.
EXERCISE 8: Using the average result from the experiments involving 10,000 points (EXERCISE 3), use your Web page to approximate PI. How accurate is the result? Is the approximation more or less accurate than the one made with only 1,000 points? What would you expect? Explain your answers.