SOLUTIONS Instructor Linda C. Stephenson SOLUTIONS 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. Everyone s plot and data columns will be different I will look at them individually! Here s what the plot should basically look like: Notice how at the beginning of the plot (red circle), with low total trials, the probability is bouncing around a lot. However, after many trials have taken place (yellow highlight), the experimental probability basically is targeting the theoretical probability. 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. Everyone s frequency tables will be different I will look at them individually! Page 1 of 5
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. See the complete sample space on the next page, where I highlighted the outcomes with a sum > 12. There are 56 outcomes with a sum > 12, and a total of 216 outcomes. Therefore: P(sum > 12) = 56 = 0. 259 216 2. Fill in the following table with your experimental probabilities (from Excel) for the given trial number: Here are mine for example yours will be different: Trial Number Probability of getting a sum > 12 1 0 10 0.2 25 0.24 50 0.2 100 0.22 500 0.252 1000 0.255 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. Also, specifically comment on how the graph displays the Law of Large Numbers. At the beginning of the plot (or at the beginning of the data table), you can see that the experimental probabilities are bouncing around, for example from P = 0 on the first trial to P = 0.2 on the tenth trial. Neither of these probabilities is very close to the expected theoretical probability of 0.259. But as the trial number gets to about 500 or so and beyond, the experimental probability is basically targeting the theoretical probability of 0.259. Therefore, the Law of Large Numbers holds true, because with MANY trials, the experimental probability gets very close to the theoretical probability. Page 2 of 5
Source: https://www.easycalculation.com/faq/2545/how_many_outcomes_in_a_sample_space_w hen.php Page 3 of 5
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. Note: you can find the theoretical probabilities by making a two-way table as follows. Listed down the first column are the possible outcomes of the first spinner (1-4). Listed across the top row are the possible outcomes of the second spinner (1-8). At the intersection of each row and column is the (positive) difference of the two spinners. 1 2 3 4 5 6 7 8 1 0 1 2 3 4 5 6 7 2 1 0 1 2 3 4 5 6 3 2 1 0 1 2 3 4 5 Note that there a total of 4x8 = 32 outcomes, all of which are equally likely. For each specific difference, take the number of ways to get that difference divided by the total number of ways, 32. 4 3 2 1 0 1 2 3 4 Difference of Numbers on Spinners Theoretical Probability in Decimal Form (do NOT round off) 0 =4/32 = 0.125 1 = 7/32 = 0.21875 2 = 6/32 = 0.1875 3 = 5/32 = 0.15625 4 = 4/32 = 0.125 5 = 3/32 = 0.09375 6 = 2/32 = 0.0625 7 = 1/32 = 0.03125 Page 4 of 5
2. 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. These will all be different, depending on your results from the applet. Here are mine, for example: Difference of Numbers on Spinners for 10 Spins for 100 Spins for 50,000 Spins 0 0.1 0.14 0.122 1 0.6 0.2 0.223 2 0.1 0.15 0.186 3 0.0 0.17 0.15 4 0.1 0.11 0.126 5 0.1 0.11 0.095 6 0.0 0.11 0.063 7 0.0 0.01 0.031 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. As with Part A it should show that as the number of trials gets very large, the experimental probability approaches the theoretical probability, which is exactly what the Law of Large Numbers says. It won t always be perfect, but in general it should be true. For example, look at my results for a difference = 2 on the two spinners. The theoretical probability of this result is 0.1875. After 10 spins, my experimental probability was 0.1, which is very far away from theoretical. After 100 spins it is 0.15, which is definitely a lot closer. Then after 50,000 spins it was 0.186, which is extremely close to the theoretical probability of 0.1875. And in general, if you look down your column of experimental probabilities after 50,000 spins, you will see that they are almost exactly equal to the theoretical probabilities. Page 5 of 5