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Example 12.16 Simulating Games of Chance

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12.112.1 | 12.2 | 12.3 | 12.4 | 12.5 | 12.6 |12.7 | 12.8 | 12.9 | 12.10 | 12.11 | 12.12 | 12.13 | 12.14 | 12.15 | 12.1712.212.312.412.512.612.712.812.9 12.1012.1112.1212.1312.1412.1512.17 Background Information n As of press time for this book, the most recent NCAA Basketball Tournament was the 1999 tournament. n You might recall that Duke was an overwhelming favorite to win – but it didnt win. n Of course, on the Sunday evening when the 64-team field was announced, we did not know which team would win.

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12.112.1 | 12.2 | 12.3 | 12.4 | 12.5 | 12.6 |12.7 | 12.8 | 12.9 | 12.10 | 12.11 | 12.12 | 12.13 | 12.14 | 12.15 | 12.1712.212.312.412.512.612.712.812.9 12.1012.1112.1212.1312.1412.1512.17 Background Information -- continued n All we knew were the pairings and the team ratings, based on Sagarins nationally syndicated rating system. n How could we use this information to simulate the entire tournament? n How accurate were simulations predictions? n How could you repeat this simulation for future tournaments?

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12.112.1 | 12.2 | 12.3 | 12.4 | 12.5 | 12.6 |12.7 | 12.8 | 12.9 | 12.10 | 12.11 | 12.12 | 12.13 | 12.14 | 12.15 | 12.1712.212.312.412.512.612.712.812.9 12.1012.1112.1212.1312.1412.1512.17 Solution n We need to make on probabilistic assumption. From the point, it is a matter of playing out the games and doing the required bookkeeping. n To understand the one probabilistic assumption, suppose team A plays team B and Sagarins ratings for these teams are 85 and 78, for example. n Then Sagarin predicts that the actual point differential in the game will be the difference between the ratings, 7. We take this one step further. n We assume the actual point differential is normally distributed with mean equal to Sagarins prediction, 7, and standard deviation 10.

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12.112.1 | 12.2 | 12.3 | 12.4 | 12.5 | 12.6 |12.7 | 12.8 | 12.9 | 12.10 | 12.11 | 12.12 | 12.13 | 12.14 | 12.15 | 12.1712.212.312.412.512.612.712.812.9 12.1012.1112.1212.1312.1412.1512.17 NCAA_99.XLS n We will only outline the simulation model. n This file contains the full details. n The file consist of six worksheets. –The first lists the teams and their Sagarin ratings. –The next four sheets simulate the results of the first four rounds of the tournament. –The last sheet simulates the semifinal and final games. n Winners from round one are automatically carried over to the next round with appropriate formulas. Because the model requires several sheets we cannot use @Risk.

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12.112.1 | 12.2 | 12.3 | 12.4 | 12.5 | 12.6 |12.7 | 12.8 | 12.9 | 12.10 | 12.11 | 12.12 | 12.13 | 12.14 | 12.15 | 12.1712.212.312.412.512.612.712.812.9 12.1012.1112.1212.1312.1412.1512.17 Developing the Model n Therefore, we go back to the methods discussed at the beginning of Chapter 11. n This requires two changes from the @Risk way of doing things. –First, we must use a combination of the NORMINV and RAND Functions to generate normally distributed random numbers. –Second, we need to use a data table to replicate the simulation – that is, play the entire tournament over many times. This data table appears on the same sheet as the final game. n Portions of this table appear on the next slide.

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12.112.1 | 12.2 | 12.3 | 12.4 | 12.5 | 12.6 |12.7 | 12.8 | 12.9 | 12.10 | 12.11 | 12.12 | 12.13 | 12.14 | 12.15 | 12.1712.212.312.412.512.612.712.812.9 12.1012.1112.1212.1312.1412.1512.17

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12.112.1 | 12.2 | 12.3 | 12.4 | 12.5 | 12.6 |12.7 | 12.8 | 12.9 | 12.10 | 12.11 | 12.12 | 12.13 | 12.14 | 12.15 | 12.1712.212.312.412.512.612.712.812.9 12.1012.1112.1212.1312.1412.1512.17 Developing the Model -- continued n We now describe the essential features of the model. –Teams and ratings. We first enter the teams and their ratings, as shown. The teams shown here were in the East region. Duke played Florida A&M in the first round, Charleston played Tulsa, and so on. –Simulate rounds. Jumping ahead to the third-round simulation shown on the next slide, we capture the winners from the previous sheet and then simulate the games in round 3. For example, the formulas in the range are =Round!F5, =VLOOKUP(B5,Ltable,2), VLOOKUP(B5,Ltable,3)-VLOOKUP(B6,Ltable,3), =NORMINV(RAND( ),D5,10), =IF(E5>0,B5,B6), and =VLOOKUP(F5,Ltable,2).

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12.112.1 | 12.2 | 12.3 | 12.4 | 12.5 | 12.6 |12.7 | 12.8 | 12.9 | 12.10 | 12.11 | 12.12 | 12.13 | 12.14 | 12.15 | 12.1712.212.312.412.512.612.712.812.9 12.1012.1112.1212.1312.1412.1512.17

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12.112.1 | 12.2 | 12.3 | 12.4 | 12.5 | 12.6 |12.7 | 12.8 | 12.9 | 12.10 | 12.11 | 12.12 | 12.13 | 12.14 | 12.15 | 12.1712.212.312.412.512.612.712.812.9 12.1012.1112.1212.1312.1412.1512.17 Developing the Model -- continued –Column B gets the indices of the second round winners from column F of the Round 2 sheet. Columns C and D get the names and differences between ratings from the original data shown. Column E simulates the point differentials in the games. Column F and G record the indices and names of the winners. –Replications. After using the logic in the previous step to play out all of the rounds, we finally get the tournament champion in cell G12. Then we create a link to this cell in cell J4 and create a data table in the range I4:J104 to replicate the tournament and record the winner 100 times.

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12.112.1 | 12.2 | 12.3 | 12.4 | 12.5 | 12.6 |12.7 | 12.8 | 12.9 | 12.10 | 12.11 | 12.12 | 12.13 | 12.14 | 12.15 | 12.1712.212.312.412.512.612.712.812.9 12.1012.1112.1212.1312.1412.1512.17 Developing the Model -- continued –Tally the winners. A nice final touch is to list all 64 teams in column L and then tally the number of replications where each team won the tournament with a COUNTIF function. Specifically the formula in M4 is =COUNTIF(Winners,=&L4) which can then be copied down column M.

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12.112.1 | 12.2 | 12.3 | 12.4 | 12.5 | 12.6 |12.7 | 12.8 | 12.9 | 12.10 | 12.11 | 12.12 | 12.13 | 12.14 | 12.15 | 12.1712.212.312.412.512.612.712.812.9 12.1012.1112.1212.1312.1412.1512.17 Results n The results are shown on the next slide. They show what a strong favorite Duke was. n Duke won the tournament in 59 of the 100 replications!. n The closest competitors were the University of Connecticut with 21 wins and Michigan State and Auburn with 5 wins each. n The majority of the 64 teams never won at all. As with any simulation, the results of this simulation depend on random number and are never definitive, but the model was certainly good at picking the best two teams in 1999.

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12.112.1 | 12.2 | 12.3 | 12.4 | 12.5 | 12.6 |12.7 | 12.8 | 12.9 | 12.10 | 12.11 | 12.12 | 12.13 | 12.14 | 12.15 | 12.1712.212.312.412.512.612.712.812.9 12.1012.1112.1212.1312.1412.1512.17

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