F +11:11 98780= UPN @ 54% K TOO O + 9.2 Analysing a Culminating Probability Project Learning Goals am learning to critique a game of chance and the related analysis Investigate Analysing a Culminating Probability Project Kaelyn and Emma presented the following information about Triple Your Chances, Double Your Counters in a culminating project report. See section 9.1 for the game instructions. Read Kaelyn and Emma's report. Then, complete the questions to critique and analyse it. Consider how you might analyse your game of chance and what elements you would include in your report. Record your ideas. Triple Your Chances, Double Your Counters: Student Analysis of the Game Theoretical Distribution of the Sum of Three Dice There are 216 total possible sums when three dice are rolled at once. 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 3 4 5 6 7 8 4 5 6 7 8 9 5 6 7 8 9 10 6 7 8 9 10 11 7 8 9 10 11 12 8 9 10 11 12 13 A 2 4 5 6 7 8 9 5 6 7 8 9 10 6 7 8 9 10 11 7 8 9 10 11 12 8 9 10 11 12 13 9 10 11 12 13 14 3 5 6 7 8 9 10 6 7 8 9 10 11 7 8 9 10 11 12 8 9 10 11 12 13 9 10 11 12 13 14 10 11 12 13 14 15 146 4 6 7 8 9 10 11 7 8 9 10 11 12 8 9 10 11 12 13 9 10 11 12 13 14 10 11 12 13 14 15 11 12 13 14 15 16 5 7 8 9 10 11 12 8 9 10 11 12 13 9 10 11 12 13 14 10 11 12 13 14 15 11 12 13 14 15 16 12 13 14 15 16 17 553 6 8 9 10 11 12 13 9 10 11 12 13 14 10 11 12 13 14 15 11 12 13 14 15 16 12 13 14 15 16 17 13 14 15 16 17 18 V + 9.2 Analysing a Culminating Probability Project . MHR 447F +11:11 98780= UPN @ 54% K TOD + Possible Probability Expected Sum Counter Payout Probability Expected Payout Sums (x P(x) x . P(x) to Player (X) P(X) X . P(X) 3 216 216 9 216 216 4 216 216 216 5 216 216 3 216 216 6 10 216 60 216 30 216 216 7 15 105 N 30 216 216 216 216 8 216 168 2 216 42 216 216 9 216 216 216 216 10 216 216 216 216 11 216 216 216 12 25 216 300 216 216 216 13 216 N 21 216 14 15 210 30 No 216 216 216 15 10 150 216 w 216 216 216 16 216 216 216 18 17 216 216 216 18 216 18 216 216 Ex.P(x) = 2268 EX- POX) = 216 = 10.5 1.9 The expected sum of three dice in this game on any given roll is 10.5, Processes 447 which means we expect the average sum of three dice will be 10.5. Connecting The expected payout shown in the table is 1.9. So, we should expect to How would this 653 pay about 1.9 counters to players for each round of play. Assume one calculation change if counter is placed on each sum in any given round. This means we would players could place collect 16 counters from players and keep an average of 14.1 or 88% of 5 counters on the V the counters, suggesting this game is not very fair to the players. board? 10 counters? between 1 and 10 counters? + 448 MHR . Chapter 9F +11:11 98780= VPN @ 53% K TO O + Theoretical Probability of Outcomes Probability (%) 10 12 14 16 18 Outcomes Notice the pattern. Sums of 3 and 18 have the least chance of occurring in the game because there is only one way to make each sum with three dice. Sums of 10 and 11 can each be made 27 ways with three dice, so they have the highest probability of occurring. We used the probability of each possible outcome to help us decide how many counters to pay out when a player landed on a sum. Sums that are highly likely pay out fewer counters, compared to sums that are less likely. Experimental Data: Sample of 10 Die Rolls Out of 120 Rounds of Play Outcomes Actual Sum Number of Balance of Counters Chosen by of Dice Player Player Counters Won for Die Roller Player Rolled Loss Win by Player (started with 100) Kaelyn and Emma 18 12 -1 101 collected 120 rounds of experimental data. 17 10 -1 102 The table shows a sample of only the 5 5 X +3 99 first 10 rounds to highlight how they 16 10 X -1 100 recorded their data. 14 14 X +2 98 8 8 Y +2 96 A 13 9 X -1 97 148 6 10 X -1 98 14 12 -1 99 553 4 X 90 V + 9.2 Analysing a Culminating Probability Project . MHR 449F +11:11 98780= VPN @ 53% K TOD + Calculating Experimental Probability for Sums To collect data for our game, we played the game for 120 rounds and recorded our data. Sum 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Number of Times Sum Occurred 0 1 2 5 7 9 12 21 21 19 8 11 1 3 0 0 To calculate the experimental probability of each sum, we used the number of times a sum occurred out of the total number of rounds played. P(9) = # of times occurred x 100 # of rounds played 10 X 100 The calculations shown are a sample = 10% of experimental probability calculations done for each of the possible sums. P(12) = # of times occurred # of rounds played x 100 =120 X 100 = 16% Experimental vs. Theoretical Probability of Each Sum 16 Experimental Probability Theoretical Probability Probability 10 11 12 13 14 15 16 17 18 Possible Outcomes A Calculating Counter Payout Based on Experimental Probability From 120 Rounds 449 Sum 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Number of Times Sum Occurred 0 1 2 5 7 9 12 21 21 19 8 11 1 3 0 0 653 Number of Matches With Player 0 1 1 2 5 6 0 0 0 0 6 7 0 2 0 0 Payout Factor 9 9 3 3 2 2 1 1 1 1 2 2 3 3 9 9 9 3 6 10 12 12 14 V Total Payout 6 After playing 120 rounds, we had to pay out a total of 72 counters. This means we paid out 0.6(2 120 counters for every round played in our game. + 450 MHR . Chapter 9F +11:11 98780= VPN @ 53% K + Conclusions The Experimental vs. Theoretical Probability of Each Sum graph shows which sums occurred more often than expected in our game. Sums of 9, 10, 11, and 12 occurred most often. Since these sums can be made in many ways with three dice, this is not a surprise. However, we would have expected sums 8, 13, and 15 to occur more often based on their theoretical probability. In the experimental distribution, sums of 3, 17, and 18 show a probability of zero, which is also not a surprise because each of these sums also has the lowest theoretical probability. The theoretical probability suggests many of the outcomes should have occurred more often, even if only a little more often. We would expect the theoretical and experimental probabilities to become more similar with more rounds of play. In reality, the game was not very fair, as we had a significant advantage for winning counters. To make the game more fair we could alter the payout structure to provide greater payouts or allow players to put counters on more than one sum. For example, provide a grouping of (3, 4, 17, 18} with a 9:1 payout ratio. If a player placed a counter on the {3,4,17,18} group, the payout would be 9 counters if one of those sums were rolled. 1. What are the key elements of Kaelyn and Emma's analysis? 2. Could a tree diagram represent the possible outcomes of their game? 3. Does their analysis fully examine the probability of winning and any pros and cons of their game? 4. Does 120 rounds of play to calculate experimental probability seem large enough? What might happen if they played 500 or 1000 rounds? 5. a) Refer to page 15 for instructions on how to create a random number generator on a graphing calculator, or download an app. Simulate the outcomes of the game using 1000 and 5000 rounds of play. b) Compare the results to the experimental probability distribution for 120 plays. What do you notice? 6. Does the title of the game Triple Your Chances, Double Your Counters describe the probability of winning this game? Why might this title A have been chosen? 7. Why do you think Kaelyn and Emma chose the payout structure they 450 did? Do you agree with the counter payout offered to the player for each possible sum? 553 8. Do the charts and graphs provide a clear and accurate representation of their game and calculations? V 9. What elements could be added or changed to more clearly explain the game, outcomes, and probability calculations? + 9.2 Analysing a Culminating Probability Project . MHR 451F +11:11 98780= VPN @ 53% K TOO 0 + Step 4: Planning Your Report 1. Are there any similarities between Kaelyn and Emma's game and your game of chance? 2. How might you represent the key elements of your analysis? 3. What graphs or charts could you use to represent data collected in your game? 4. What calculations could you include in your game of chance analysis? How will you show these calculations? 5. Would you make any adjustments to the payout structure in your game to ensure you win, or to make it more enticing for the player to continue playing? 6. What amount of data would you collect for your game of chance? 7. Could you use a random number generator to represent your data? If so, how many plays might you simulate? What results would you expect compared to the theoretical probability of winning your game, and to your experimental distribution if you simulated more rounds of play? 8. What would be an effective title for marketing your game of chance? 9. What elements of Kaelyn and Emma's game and analysis would you keep, change, or add to make a better game and report for your culminating probability project? Step 5: Writing Your Report Project Prep A well-written report is organized, logical, and includes the necessary Refer to section information with appropriate headings. Pull all the pieces together to 9.5 for tools to tell the story of your game and its analysis in the form of a report. Your self-assess your report should include probability report. . a description of the game, rules, and required materials A . data from the game day or simulation 451 . calculations of the theoretical distribution, experimental distribution, and expected value 653 comparison of the theoretical and experimental distributions summary of conclusions, including discrepancies in the results and V insights and reflections about the game + 452 MHR . Chapter 9F +11:11 98780= VPN @ 53% K TOO O + Step 6: Finalizing Your Project Look back at the theoretical probability for the game Triple Your Chances, Double Your Counters. Were the results very likely, given the number of trials Kaelyn and Emma ran? While the game was being played, Logan was overhead saying, "This game is so easy! I could win five times in just five rounds! All I have to do is put my counters on 10!" Kaelyn and Emma did not agree and did the following calculations to disprove the statement. Since a sum of 10 has the highest probability of occurring, we assumed that Logan will place his counter on 10 for all five plays. We can use a binomial distribution to represent the number of wins that occur out of five games because: . Each roll of the dice is an independent event. . There is a fixed number of dice being rolled. . There are two possible outcomes: success or failure (either the sum on the dice is 10 or it is not). The chances of success remain constant, assuming Logan places his counter on 10 for all five rounds. The table shows the chances of winning on each of the five rounds. Number of Wins Probability of Winning Out of Out of Five Rounds Five Rounds, P(x) = ,C.p*q-x) P(x) = ,C (0.125) (0.875)5-00 Theoretical probability = 0.512 91 of rolling a sum of 0.366 36 10 = 216 N N 0.104 68 = 0.125. 0.014 95 0.001 068 0.000 030 5 Our calculations show that Logan has a 51.3% chance of winning 0 out of A 5 rounds played, assuming the same outcome is chosen for each of the five rounds. 152 Also, Logan's chance of winning 1, 2, 3, 4, or all 5 times out of five rounds gets progressively lower, suggesting that our game is actually not very easy to win. 553 V + 9.2 Analysing a Culminating Probability Project . MHR 453F +11:12 9878A= VPN @ 53% K TODD + 1. Was a binomial distribution an appropriate choice to show the chances of winning from 0-5 times in five rounds? 2. Under what conditions does a binomial distribution model not work for this game? 3. Would another distribution model fit the data from this game better? Explain how you know. 4. What kind of distribution model might best fit the data from your game of chance? Include this information in your report and justify your calculations. Step 7: Assessing Your Probability Project Use the checklist to ensure your probability project is on track. The Report Yes No Does your report include: Title page Appropriate use of headings . Description of the game and relevant background information . Outline of the rules, including how to win . Explanation of the fee structure (cost to play) . Data table representing actual data from the game day . Actual winnings earned in the game based on your data table Have you calculated and/or graphed: . Theoretical distribution (possible outcomes vs. probability) . Experimental distribution (actual outcomes vs. probability) . Expected value of the game (profit based on your theoretical calculations) Does your analysis of the results: . Compare and contrast the theoretical and actual distributions . Discuss any discrepancies in the results . Provide insights and reflections about the game . Summarize your conclusions A Review the following elements of your report: . Terminology: Are mathematical terms used correctly? . Neatness: Is your font clear and readable? Does your report look 153 professional? . Writing Skills: Have you followed proper writing conventions? 553 . Organization: Are key ideas presented logically, clearly, and concisely? Is your report focused and on topic? V . Creativity: Have you included pictures or diagrams that make the report interesting? . Technology Skills: Are your visuals easily understood? Are your graphs clearly labelled? + 454 MHR . Chapter 9