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MonteCarlo
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Intro Intro to MonteCarlo.xls This workbook introduces Monte Carlo Simulation with a simple example. In this book, we use Microsoft Excel to simulate chance processes. Typically, we use Excel to draw a sample, then compute a sample statistic, e.g., the sample average. We can repeat this process, again and again, keeping track of the result each time. We then look at the empirical histogram and summary statistics of the results in order to discover or illustrate the statistical properties of the chance process. This method of studying a chance process is called Monte Carlo simulation. The OneFreeThrow sheet shows how Excel can be used draw a random number and simulate a free throw. The Sample sheet has the outcome of 100 free throw attempts. There are other hidden sheet in this workbook. OneFreeThrow To simulate a free throw, we use a random draw in an IF statement. Scroll down to see how. The random draw uses the formula, "=RANDOM()", to get a uniformly distributed number in the 0,1 interval. Click in cell B5 to see the formula. Hit F9 (or CTRL - =) to draw a new number. For more on random number generation, see sections 9.2 and 9.3. Random draw 0.822815617463931 The RANDOM() function is the foundation of the Monte Carlo Simulation because it's our roulette wheel, but we need one more concept to complete the simulation. We need a way to tell Excel that Larry has a 90% chance of making the free throw. Fortunately, this is easily done with an IF statement. Take a look at the formulas in cells B18, D18, and D18. Cell D18 says, "if the value in cell B18 is less than 0.9, then the value of this cell is 1, else the value is 0." The structure is "expression to be evaluated," "value if true," "value if false." Notice the difference between cells D18 and D19--yet they end up doing the same thing. While you are at it, try to figure out what's going on in cells C18 and C19. Random draw 0.012194132438020 Make 1 Make 1 Combining the RAND and IF statements into a single formula, we get a virtual 90% free throw shooter: Larry Bird Simulated 1 Notice that this outcome will occasionally differ from the one above. The reason is that it's based on a different draw from the uniform distribution. Hit F9 (or CTRL - =) repeatedly (say 30 times in a row) to see how Larry's doing. As long as the value stays at 1, Larry hit the free throw. When it changes to 0, he missed. Question: "I noticed that sometimes he misses in cell B23, but makes it in cells C18 through D19. What's going on?" Answer: Excel is drawing different random numbers in cells B18 and B23. Every time it sees RANDOM() in a cell, Excel draws a new random number. Question: "Sometimes I see the word "Calculate" at the bottom of the screen. What does that mean?" Answer: Excel can be set up to automatically calculate the values of every formula whenever the user enters new information anywhere in the workbook OR it can be in Manual Calculate mode, which means that it will not automatically recalculate. Manual Calculation is needed for Monte Carlo simulation to prevent constant redrawing of random numbers. The "Calculate" message in the Status Bar at the bottom of the screen alerts the reader that Excel has not automatically recalculated the cells in the workbook. If you hit F9, the "Calculate" message will disappear. Section 9.3 explains about Calculation in more detail and shows how to set the calculation mode. Sample Percentage Made 90% This sheet contains a single repetition of 100 draws (free throw attempts). Percentage of First Ten Made 90% Attempt Result Larry Bird hits 90% of his free throws. The formula in Cell B4 informs Excel of this chance process. 1 1 The exact same formula is used in the 100 cells from B4 to B103. Click on a few of these cells to see for yourself. 2 1 Hit F9 to simulate another 100 free throws. 3 1 How did Larry do this time? 4 1 5 1 Hit F9 again and again. Watch cell B1. 6 1 How much spread or bounce is there? 7 0 8 1 Wouldn't it make sense to keep track of Larry's sample percentage? 9 1 Click on the button below to do so. 10 1 11 1 12 1 13 1 Curious about streaks? Click on the button below. 14 1 Click this button several times. 15 1 16 1 17 0 Do you think Max Streaks follow a normal distribution? 18 1 Click on the button below to see a Monte Carlo simulation that answers this question. 19 1 20 1 21 1 22 1 23 1 24 1 25 1 26 1 27 1 28 1 29 1 30 1 31 1 32 1 33 1 34 0 35 0 36 1 37 1 38 0 39 1 40 1 41 1 42 1 43 1 44 1 45 1 46 1 47 1 48 1 49 1 50 1 51 1 52 1 53 1 54 1 55 1 56 1 57 1 58 1 59 1 60 1 61 1 62 1 63 1 64 1 65 0 66 1 67 1 68 1 69 1 70 1 71 1 72 1 73 1 74 1 75 1 76 1 77 1 78 0 79 1 80 1 81 0 82 1 83 0 84 1 85 1 86 1 87 1 88 1 89 1 90 1 91 1 92 1 93 1 94 1 95 1 96 1 97 1 98 1 99 0 100 1 Monte Carlo Simulation FT% Streak Finder Monte Carlo Simulation Max Streak Q&A Q&A for MonteCarlo.xls 1) Run a Monte Carlo with 1,000 repetitions of the percentage made of 100 free throws. Use the Take a Picture button to take a picture of your results. Copy and paste the picture in a Word document. 2) Run a Monte Carlo with 10,000 repetitions of the percentage made of 100 free throws. Take a picture, copy it, and paste it in the Word document below the 1,000 repetition Monte Carlo. 3) In what sense is the 10,000 repetition Monte Carlo better than the 1,000 repetition Monte Carlo in this example? 4) Run a Max Streak Monte Carlo. Take a picture, copy it, and paste it in the Word document. 5) What are the chances that the virtual Larry Bird gets a max streak of 51 in a row or more? Explain how you arrived at your answer. MCSim Monte Carlo Simulation of Sample Percentage with 1000 Repetitions Min = 80% Sample Number Sample Percentage Max = 97% Bin Value Number of Samples 1 95% Excel Tip: Pause the cursor over one of the bars, the number of samples (or height of the bar) is displayed. Average = 89.91% 78% 0 2 93% Time Elapsed 2 secs SD = 3.012% 79% 0 3 92% 80% 2 4 89% 81% 3 5 93% 82% 7 6 93% 83% 7 7 84% 84% 29 8 89% 85% 37 9 92% 86% 48 10 84% 87% 63 11 91% 88% 111 12 89% 89% 128 13 87% 90% 124 14 88% 91% 131 15 94% 92% 103 16 89% 93% 92 17 96% 94% 66 18 89% 95% 28 19 88% 96% 15 20 94% 97% 6 21 92% 98% 0 22 90% 99% 0 23 90% 100% 0 24 93% 25 87% 26 93% 27 92% 28 90% 29 87% 30 86% 31 86% 32 90% 33 92% 34 91% 35 92% 36 90% 37 86% 38 89% 39 89% 40 97% 41 88% 42 92% 43 94% 44 93% 45 87% 46 91% 47 90% 48 94% 49 92% 50 92% 51 92% 52 91% 53 90% 54 91% 55 91% 56 94% 57 87% 58 84% 59 89% 60 91% 61 90% 62 88% 63 91% 64 91% 65 88% 66 92% 67 84% 68 89% 69 94% 70 91% 71 93% 72 90% 73 89% 74 90% 75 91% 76 91% 77 89% 78 91% 79 89% 80 94% 81 93% 82 91% 83 96% 84 88% 85 92% 86 90% 87 89% 88 88% 89 84% 90 92% 91 85% 92 88% 93 91% 94 93% 95 93% 96 91% 97 87% 98 94% 99 88% 100 89% 101 91% 102 91% 103 91% 104 83% 105 88% 106 90% 107 88% 108 93% 109 84% 110 86% 111 89% 112 88% 113 86% 114 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91% 990 91% 991 93% 992 91% 993 94% 994 91% 995 89% 996 88% 997 93% 998 94% 999 88% 1000 91% MCSim Min = 80% Average = 89.91% SD = 3.012% Summary Statistics Max = 97% %Made Empirical Histogram for 1000 Repetitions Streak Monte Carlo Simulation of Sample Percentage with 10000 Repetitions Min = 8 Sample Number Max Sample Streak Max = 89 Bin Value Number of Samples 1 15 Average = 27.0 0 to 5 0 2 22 Time Elapsed 2.00 secs SD = 9.7 6 to 10 13 3 32 11 to 15 640 4 16 16 to 20 2084 5 21 21 to 25 2466 6 43 26 to 30 1913 7 17 31 to 35 1206 8 16 36 to 40 722 9 22 41 to 45 443 10 27 46 to 50 230 11 38 51 to 55 143 12 28 56 to 60 69 13 36 61 to 65 39 14 36 66 to 70 16 15 29 71 to 75 7 16 21 76 to 80 5 17 25 81 to 85 3 18 14 86 to 90 1 19 16 91 to 95 0 20 19 96 to 100 0 21 18 22 27 23 25 24 26 25 31 26 20 27 21 28 29 29 24 30 26 31 34 32 23 33 17 34 24 35 26 36 21 37 18 38 15 39 25 40 23 41 24 42 33 43 43 44 56 45 23 46 33 47 23 48 31 49 23 50 19 51 17 52 34 53 17 54 26 55 20 56 29 57 34 58 33 59 33 60 35 61 56 62 29 63 23 64 21 65 29 66 30 67 19 68 32 69 19 70 39 71 29 72 16 73 31 74 28 75 26 <= 0 to to to to to to to to Min = 8 Min = 8 to to to to to to to to to 0 0 0 76 13 77 14 78 21 79 32 80 23 81 25 82 43 83 16 84 23 85 29 86 23 87 17 88 20 89 34 90 19 91 22 92 17 93 28 94 17 95 17 96 18 97 24 98 18 99 17 100 18 101 22 102 40 103 26 104 38 105 18 106 53 107 22 108 21 109 20 110 20 111 25 112 16 113 20 114 41 115 64 116 28 117 16 118 23 119 22 120 34 121 14 122 32 123 24 124 15 125 23 126 30 127 20 128 33 129 28 130 22 131 19 132 28 133 23 134 21 135 29 136 27 137 22 138 28 139 21 140 36 141 16 142 16 143 39 144 22 145 55 146 28 147 30 148 19 149 19 150 28 151 32 152 44 153 16 154 15 155 23 156 36 157 42 158 24 159 29 160 35 161 21 162 61 163 19 164 29 165 19 166 18 167 32 168 24 169 28 170 18 171 34 172 40 173 18 174 36 175 22 176 31 177 17 178 19 179 21 180 14 181 15 182 72 183 23 184 22 185 32 186 18 187 36 188 27 189 44 190 36 191 41 192 16 193 29 194 50 195 16 196 33 197 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