Assignment 5 due Monday, May 7

Transcription

1 due Monday, May 7 Simulations and the Law of Large Numbers Overview In both parts of the assignment, you will be calculating a theoretical probability for a certain procedure. In other words, this uses the Classical Method discussed on p. 251 of the book (p. 4 of the Chapter 5 Handouts). Then you will simulate the procedure on the computer, and calculate the experimental probabilities using the Empirical Approach. This is discussed on p. 3 of the Chapter 5 Handouts. In both cases, you will investigate whether the Law of Large Numbers applies as the number of experiments becomes very large. Part A Getting a sum > 12 when rolling three 6-sided dice In Part A of the assignment, you re going to use STATDISK to simulate rolling three 6-sided dice, 1000 times. STATDISK will add the numbers on the three dice together. The event that you are looking for is getting a sum of the numbers on the three dice that is GREATER THAN 12. For example, the sum shown on these three dice is 13. After each roll of the dice, you will calculate your experimental probability of getting a sum > 12, using the relative frequency approximation of probability. Then you will plot the experimental probability as a function of trial # for trials 1 through You will compare the experimental probability to the theoretical probability, and see what happens as the number of trials in the experiment gets very large. Tip: you will probably want to look up the sample space for rolling three dice in order to calculate the theoretical probability of getting a sum > 12. Note: I am running Excel 2013 on my home computer. If you are running a different version, then my instructions below may not correspond exactly to your version. However in all cases it should be basically the same, so just do your best to figure out any modifications for your version of Excel. Page 1 of 14

2 Part A Procedure 1. Start Excel, and start with a new (blank) file. The first thing you want to do is create a column of numbers 1, 2, 3, 4,, 999, 1000 in column A. Here s a handy way to do that: Note: to enter something into a cell in Excel, click on the cell, type whatever you want (a number, or a formula), then always hit Enter. a. Enter the numbers 1, then 2, then 3 in the first three cells of column A. b. Click on cell A1 and, while holding the mouse button down, drag the mouse downward so that cells A1, A2, and A3 are highlighted, then release the mouse button. c. Now use the mouse to click and drag the lower right corner of cell A3 downward so that all of the cells down to A1000 are highlighted. Column A should now contain 1, 2, 3,, 999, 1000 in cells A1 through A Now open STATDISK. You are going to simulate rolling 3 six-sided dice a total of 1000 times. Along the top, go to Data/Dice Generator. Fill in the boxes for Sample Size 1000, Number of Dice 3 and Number of Sides 6, then hit Generate. You will see two columns, the Roll number , and the Total, which is the sum of the three dice. Hit the Copy button, which will ONLY copy the Total column. Now you re going to paste that column of totals into column B of your Excel file. Click on cell B1 again and hit Ctrl-V (to paste). You should now have an entire column of sums, representing the results of one thousand rolls of a three dice. See my screenshot of the first 15 rows: 3. Next, move to column C in Excel. In this column, you are going to calculate a running total of how many of the sums are greater than 12. Click on cell C1, and enter the following EXACTLY: =COUNTIF($B$1:B1,">12") Then hit enter. This command is telling Excel to update (increment) the count if you have a number that is greater than 12. In other words, if your sum was a 13, 14, 15, 16, 17 or 18, it will increment the count by 1. Page 2 of 14

3 So, if you happen to have a sum of > 12 in cell B1, then you will get a 1 in cell C1. Otherwise you will get a 0 in cell C1. In total, what this command will do is to count up how many numbers > 12 appear starting at cell B1 and continuing through the specified range. So it is a cumulative total of the successful rolls. So far, my range just starts and ends at B1, so there s not much to count up yet. Now you re going to copy and paste the formula in cell C1 to the rest of the cells in column C. Click on cell C1 and hit Ctrl-C (to copy). Then click on cell C2, and while you re clicking drag the mouse all the way down to cell C1000. Release the mouse and hit Ctrl-V (to paste). Here s my screenshot of the first 15 rows nows: Notice how every time I got another sum > 12, the cumulative total in column C is incremented up by one. That s what column C is doing: keeping track of the total number of sums > 12. Note: Check to make sure it is working correctly! 4. Okay, the last column you are going to calculate will be your empirical/experimental probability of getting a sum > 12. As defined on p. 3 of the Chapter 5 Handouts, this probability is: P(A) = Frequency of event Number of trials So in other words, you need to calculate the cumulative total of sums > 12 in column C divided by the number of trials in column A. To do this, Click on cell D1 and type: =C1/A1 (*see note in bold below) *Note: instead of actually typing the C1 and A1, you can just click on the cells themselves, and the cell references will be entered into the formula. Then hit enter. Copy and paste this formula to the rest of the cells in column D, just like you did in step 3. Page 3 of 14

4 Here s my screenshot now: Note: You will either have a 1 or a 0 in cell D1 depending on whether you got a sum > 12 (or not) in cell B1. 5. Now you re going to make a plot of the experimental probability (column D) as a function of the trial number (column A). This is going to be a scatter plot. Start by clicking in any empty cell near the top, so that you do not have any data currently selected. Along the top, go to the Insert Tab, choose the down arrow next to the scatter plot icon, and select the option in the second row, first column (Scatter with Straight Lines and Markers). An empty Chart box should appear in your worksheet. You should also now be on the Design Tab, so hit Select Data (upper right). Under Legend Entries (Series), hit Add. Under Series X values, click the little red arrow box and use your mouse to select the entire column A (cells A1 through A1000). Click the red arrow box again to return. Under Series Y values, click the little red arrow box and use your mouse to select the entire column D (cells D1 through D1000). Click the red arrow box again to return. Hit OK, OK. In other words, you are plotting column D as a function of column A, so each ordered pair is this: (trial number, probability). Page 4 of 14

5 At this point, you should have a rough plot which looks kind of like this: (although, if you DID get a sum > 12 on your first roll, your graph will start up at 1.0 on the y-axis). Use the following instructions to add a chart title, x-axis label and y-axis label: Chart Title: Experimental Probability of getting a sum > 12 on 3 dice X axis label: Trial Number Y axis label: Probability Click on the plot area, and go to the Design tab at the top right of the worksheet. Then at the upper left you should see the Add Chart Element box, as shown here: Click on the down arrow, and you will see options to add axis titles and a chart title. A couple of other minor improvements: Change the scale of the x-axis so that it goes from 0 through 1000 (instead of 1200). Right click on the x-axis to activate it, it should be outlined in a box now. Left click, and choose Format Axis. Change the maximum value to Here s what my plot looks like now. Remember these are RANDOM numbers, so yours won t look exactly like mine (especially at the beginning). Page 5 of 14

6 That s all the calculations you have to do for this portion of the assignment! See below under What you Need to Turn In. Page 6 of 14

7 Part B Spin the Spinner In Part B of the assignment, you re going to use an online applet to simulate spinning two spinners 10, 100 and 50,000 times. When you first go to the link below, you will see the following spinner set-up: Do exactly this, and in this order: 1. Hit the Clear button (both spinners will disappear) 2. Hit New spinner 3. Change the number in the box to 8 4. Hit New spinner (again) 5. Hit the sum/difference box in the upper right Now the spinner set-up should look like this, with one spinner 1-4 and one 1-8: This is the set-up that you need for this portion of the assignment, so make sure that you do this first! Make sure that it is recording the DIFFERENCE of the spinners (not the sum). Note: I think people with Macs may have trouble with this applet, you may need to use a different computer. Page 7 of 14

8 Each time you play, both spinners will randomly spin and each will land on a number: 1 4, or 1 8. The difference of the two numbers on the spinners will be calculated. It will always record the non-negative difference, in other words the absolute value. For example, if you get a 4 and an 8, the difference will be 4, never 4. Before you run the simulation, you should calculate the theoretical probabilities of each difference that is possible, that is, the differences of the two numbers on the spinners. The possible differences are: 0, 1, 2, 3, 4, 5, 6, and 7. You are using the Classical Method for this. Hint you could use a two-way table to help calculate these probabilities. Fill in the table on the Answer Sheet with the theoretical probabilities. Then you will run the actual simulation. Go to the following link: (also the link is given on the Assignment page) Note: I tested this in Internet Explorer, Mozilla and Google Chrome, and it worked in all three. If you get a weird error message in one browser, then try a different one. Or like I said above, if it doesn t work on your Mac, then try a different computer. You will run three different cases: 10 spins, 100 spins and 50,000 spins. To do the 10 spin case, you will have to hit the Run Once button 10 times. Each time, it will update the bar chart on the right hand side of the screen, and will update the number of spins in the lower right corner: Also, you can click on the Relative frequency table line (bottom right), and it will hide the bar chart and show the tabular data instead, so you can toggle back and forth between those if you want. (see screenshot next page) Page 8 of 14

9 The column on the left shows the differences that were obtained (the values 0 through 7), and the column on the right is the relative frequency in decimal form for each sum. Notice that there are other values included in the table (on up to 60 if you scroll down), but those do not apply for this procedure. So the applet has already taken the counts for each outcome and divided by the number of trials, so it has calculated the relative frequency for you. These relative frequencies are your experimental probabilities for each trial, because it represents the number of times that sum was obtained divided by the total number of trials. Page 9 of 14

10 For each of your three trials, I want you to print out the relative frequency table showing the relative frequencies and the number of spins at the bottom. This is what I want to see : Use the snipping tool to print these out, or do a screen print and then crop it. You will have a total of three of these to print out: 10 spins, 100 spins, and 50,000 spins. Make sure you hit the Clear button (on the right-hand side) after you do one trial (say of 10 spins) and before you start the next, otherwise it will add your new trials to the ones you already did. For example, you would end up with 110 spins instead of 100. Also, make sure that you go back and look at the bar chart each time, as it gives a good visual of what is happening as you do more trials. Actually it is interesting to watch the shape change if you just keep hitting the Run x 100 button several times. Page 10 of 14

11 What You Need to Turn In Part A Getting a sum > 12 when rolling three 6-sided dice 1. Printout of your plot from Excel. 2. Printout of the first 50 lines of your four data columns from Excel. Do NOT print out the whole thing! I do not want to see 1000 lines of data. You can select a range of cells that you want to print using the mouse, go to the Page Layout tab, then click Print Area/Set Print Area. 3. Filled out Answer Sheet (1 page). Part B Spin the Spinner 1. Printout of your Relative frequency table (see previous page) from the applet screen for each trial: 10, 100 and 50,000 spins. Make sure that you include the number of spins at the bottom of the printout. So, three total tables to print out. 2. Filled out Answer Sheet (2 pages). REMINDER Everyone needs to do their own Excel spreadsheet, and their own simulation on the applet! Page 11 of 14

12 Name: Answer Sheet Part A Getting a sum > 12 when rolling three 6-sided dice 1. What is the theoretical probability of getting a sum > 12 when rolling three 6-sided dice, using the Classical Method? Reduced fraction or decimal to three significant figures. 2. Fill in the following table with your experimental probabilities (from Excel) for the given trial number: Trial Number Experimental Probability of getting a sum > Briefly discuss how this simulation, including your results for questions (1) and (2), relates to the Law of Large Numbers. Give specific results as an example. Also, specifically comment on how the graph displays the Law of Large Numbers. Page 12 of 14

13 Name: Part B Spin the Spinners 1. Calculate the theoretical probability of getting each of the following differences when the two spinners are spun and the difference of the two numbers is found, using the Classical Method. Difference of Numbers on Spinners 0 Theoretical Probability in Decimal Form (do NOT round off) List the experimental probabilities that you observed for the following number of spins. Remember, that is the same as the relative frequencies. Do not round off the values. Difference of Numbers on Spinners Experimental Probability of Each Difference for 10 Spins Experimental Probability of Each Difference for 100 Spins Experimental Probability of Each Difference for 50,000 Spins Page 13 of 14

14 Name: 3. Briefly discuss how this simulation, including your results for questions (1) and (2), relates to the Law of Large Numbers. Give specific results as an example. Page 14 of 14