Part A Getting a 3 or less on a single roll of a 10-Sided Die 1. Printout of your plot from Excel. 2. Printout of the first 50 lines of your four data columns. 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 the experimental probabilities settle in towards the end of the plot, right around a value of 0.4, which is 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!
Answer Sheet Part A Getting a 3 or less on a single roll of a 10-Sided Die 1. What is the theoretical probability of rolling a 3 or less on a single roll of a 10-sided die, using the Classical Method? Reduced fraction or decimal to three significant figures. There are 4 ways to get a 3 or less out of a total of 10 outcomes (0, 1, 2, or 3), therefore: P( 3 or less ) = 4 10 In decimal form: P( 3 or less ) = 0.4 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 Experimental Probability of getting a 3 or less 1 0 10 0.4 25 0.4 50 0.48 100 0.47 500 0.418 1000 0.405 3. Briefly discuss how this simulation, including your results for questions (1) and (2), relates to the Law of Large Numbers. It shows 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. 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 = 0.48 on the 50th trial. Neither of these probabilities is very close to the expected theoretical probability of 0.4. But as the trial number gets to about 500 or so and beyond, the experimental probability is basically targeting the theoretical probability of 0.4. You can see this clearly on the graph, that the experimental probabilities are bouncing around at the beginning of the graph, and then they really settle in and target the theoretical probability after a large number of trials.
Part B Spin the Spinners 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. Listed across the top row are the possible outcomes of the second spinner. At the intersection of each row and column is the difference of the two spinners. Notice that there are 49 total outcomes. 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 2 3 4 5 6 7 8 9 3 4 5 6 7 8 9 10 4 5 6 7 8 9 10 11 5 6 7 8 9 10 11 12 6 7 8 9 10 11 12 13 7 8 9 10 11 12 13 14 Use the table to count up the number of ways that each outcome can occur. Sum of Two Spinners Number of ways that sum can occur P(sum) = Number of ways / 49 (49 = total no. of outcomes) 2 1 = 1/49 = 0.0204 3 2 = 2/49 = 0.0408 4 3 (etc.) = 0.0612 5 4 0.0816 6 5 0.1020 7 6 0.1224 8 7 0.1429 9 6 0.1224 10 5 0.1020 11 4 0.0816 12 3 0.0612 13 2 0.0408 14 1 0.0204
1. Calculate the theoretical probability of getting each of the following sums when the two spinners are spun and the sum of the two numbers is found, using the Classical Method. Sum of Numbers on Spinners Theoretical Probability in Decimal Form (to 4 decimal places) 2 0.0204 3 0.0408 4 0.0612 5 0.0816 6 0.1020 7 0.1224 8 0.1429 9 0.1224 10 0.1020 11 0.0816 12 0.0612 13 0.0408 14 0.0204 (continued on next page)
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: Sum of Numbers on Spinners Experimental Probability of Each Sum for 10 Spins Experimental Probability of Each Sum for 100 Spins Experimental Probability of Each Sum for 50,000 Spins THEORETICAL (for comparison) 2 0 0.02 0.02 0.0204 3 0 0.04 0.039 0.0408 4 0 0.05 0.06 0.0612 5 0.1 0.12 0.081 0.0816 6 0.2 0.14 0.105 0.1020 7 0.2 0.13 0.123 0.1224 8 0 0.13 0.146 0.1429 9 0.3 0.08 0.119 0.1224 10 0.1 0.07 0.101 0.1020 11 0 0.09 0.081 0.0816 12 0 0.05 0.059 0.0612 13 0.1 0.04 0.04 0.0408 14 0 0.04 0.019 0.0204
3. Briefly discuss how this simulation, including your results for question (1) and (2), relates to the Law of Large Numbers. 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 sum = 11 on the two spinners. The theoretical probability of this result is 0.0816. After 10 spins, my experimental probability was 0, not close at all. After 100 spins it is 0.09, closer by too high now. Then after 50,000 spins it was 0.081, almost exactly equal to the theoretical probability. In general, this pattern will hold true for all of the results. You should have noticed that your 50,000 spin probabilities were VERY close to the predicted theoretical probabilities.