'Fans 9%]6 9.2 Analysing a culminating Probability Project 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 possible sums when three dice are rolled at once. l 2 3 I 3 4 3 7 B 8 9 9 10 ll 12 13 4 789 8 9 T5F6113 9EJBEI The expected sum of three dice in this game on any given roll is ELL, which means we expect the average sum of three dice will be &, The expected payout shown in the table is Ji So, we should expect to pay about counters to players for each round of play. Assume one counter is placed on each sum in any given round. This means we would collect 11: counters From players and keep an average of 14.1 or 887. of the counters, g esting this game is the players. 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, T +6:13 9A6:A- VPN @ 89% K TO + Experimental Data: Sample of 10 Die Rolls Out of 120 Rounds of Play Kaelyn and Emma collected 120 rounds of experimental data. The table shows a sample of only the first 10 rounds to highlight how they recorded their data 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) 18 12 X - 10 17 10 X -1 102 5 5 X +3 99 16 10 X -1 100 14 14 X +2 98 8 8 X +2 96 13 9 X -1 97 6 10 X -1 98 14 12 X -1 99 19 90 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 Experimental vs. Theoretical Probability of Each Sum Experimental Probability Theoretical Probability 12- Probability V I w