What is the expected number of rolls to get a Yahtzee?

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1 Honors Precalculus The Yahtzee Problem Name Bolognese Period A Yahtzee is rolling 5 of the same kind with 5 dice. The five dice are put into a cup and poured out all at once. Matching dice are kept out and the non-matching dice are put back into the cup and rolled again. In a traditional game of Yahtzee, a maximum of 3 rolls are permitted. However, our essential question is: What is the expected number of rolls to get a Yahtzee? Part I: (Experimental Probability) 1) Using your intuition, how many rolls would you expect would be needed to get a Yahtzee? Explain your reasoning. 2) Now let s try it! With a partner, roll 5 dice in a cup. Reserve any that match and keep rolling the leftover die until you reach a Yahtzee (all five dice have the same kind). In the table below, write the total number of rolls required to reach a Yahtzee in each trial. Do this for TEN TRIALS. Trial # Rolls for a Yahtzee 3) On the last page, complete the frequency chart for YOUR group data. Then copy the class data once completed. This class data will be imported into an Nspire file for you to work with in the following steps. COMPLETE ONCE YOU HAVE THE CLASS NSPIRE FILE: 4) a) To view a graph of the class frequency data, you will need to select both columns. Hit the up arrow enough to highlight one column. Holding down the SHIFT key, hit the right arrow to highlight the next column. b) With these columns highlighted, hit CTRL then the menu button. You will see a pop up menu. Select 8. Summary Plot. Change the settings as shown below to display on a new page. c) To change between plot types, hit menu, 1. Plot Type, and select either 2. Box Plot or 3. Histogram. Also, try toying around with the settings by going to menu, 2. Plot Properties for either type of graph.

2 5) Using your calculator, graph the class frequency data. Represent the data using 2 different types of graphs available on the Nspire. Sketch these graphs below and label the important aspects of each graph. Using these graphs, name at least 5 characteristics about the distribution. Try to include discussion about shape, measures of center, spread, etc. 6) Experimentally, what seems to be the expected number of rolls necessary to obtain a Yahtzee? Justify your answer. How does this answer compare to the Yahtzee java simulation? Part II: (Theoretical Probability) 7) Using counting techniques, fill in the chart below to find the various possibilities on the first roll: Results of the First Roll Method of Computation # of Possibilities All 5 match 3 match-1 not-1 not 2 match-1 not-1 not-1 not 5C C 2 3 C 2 6 C 2 4 Total =

3 8) Rather than summing cases, show how the total number of outcomes be computed in a different way. 9) Complete the probability table for the maximum number of common die on the first roll. Maximum # Matches Probability (as a fraction and as a decimal) 10) The table below is called a transition matrix, whose entries are probabilities from one state to another. For example, the entry in the 3 rd row and 4 th column represents the probability of having 3 matches and going to 4 matches on the next trial. From\To a) Note that some of the probabilities given in the table are zeroes. Explain why this is so. b) Let s compute the probability in the 3 rd row, 4 th column. If 3 already match, we will roll 2 die. Of the 2 die, there are 2 C 1 ways to get an additional match. The probability of a matching face is 1/6 and the probability of not getting a match is 5/6, so the total probability is 2 C = Complete the rest of the table above using similar counting techniques.

4 c) In the space below, construct a labeled tree diagram to show all the possibilities of the number of matches obtained in two rolls. Then use this tree diagram to calculate the total probability of obtaining Yahtzee in two or less rolls. d) Challenge: Determine the total probability of obtaining a Yahtzee in at most three rolls. What about n rolls?

5 3) Continued - Frequency Charts # Rolls to get a Yahtzee Frequency (Your Group) Frequency (Class)

6 Cognitive Demands: Part I: Procedures with Connections: Data is represented in multiple ways. Making connections among multiple representations helps to develop meaning. Part II: Procedures with Connections: Requires cognitive effort. General procedures cannot be followed mindlessly. Students need to engage with conceptual ideas that underlie the procedures in order to successfully complete the task. Standards for Mathematical Practice: Part I: #1 Make sense of problems and persevere in solving. #2 Reason quantitatively. #4 Model with mathematics. #5 Use appropriate tools (calculator and dice). #6 Express regularity in repeated reasoning (answer the question). Part II: #1 Make sense of problems and persevere in solving. #2 Reason abstractly and quantitatively. #6 Attend to precision. (accurate counting methods). #8 Express regularity in repeated reasoning. Standards for Mathematical Content (Stats and Probability) Part I: S-ID Summarize, represent, and interpret data on a single count or measurement variable. (1,2,3,5,6) S-IC Understand and evaluate random processes underlying statistical experiments. (1, 4, 6) Part II: S-CP Use the rules of probability to compute probabilities of compound events in a uniform probability model. (9) S-MD Calculate expected values and use them to solve problems. (2,3) GAISE: I Formulate Question: Part I (A); Part II (B) II Collect Data: Part I (A); Part II (A) III Analyze Data: Part I (B); Part II (B) IV Interpret Results: Part I (B): Part II (B)