# What are the chances?

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1 What are the chances? Student Worksheet TI-Nspire Investigation Student 90 min Introduction In probability, we often look at likelihood of events that are influenced by chance. Consider the event of rolling a pair of six-sided dice and observing the sum of the two numbers that turn up. We can observe what might happen if we roll the dice a fixed number of times (the experimental probability), and then we can use mathematics to look at what we might expect to happen in the long run (the theoretical probability). Part 1: Simulating chance events Most calculators also have a random integer command, which generates random integers between two limiting values. For example, the command randint(1,20) on the TI-Nspire CAS will generate a random integer between 1 and 20 (inclusive). To do this on the TI-Nspire CAS Press HOME-1 to create a new document, and then press 1 to add a Calculator page. Press MENU-5 to select Probability, then press 4 (Random) and then 2 (Integer). Type randint(1,20) and then press to display a random integer between 1 and 20. Press ENTER a few times to generate some extra random numbers. A number of random integers will be generated. To begin with, we will simulate the process of rolling of a single die. Question 1. What command would simulate this experiment? randint(1, 6) Question 2. Now try the command 60 times, and record the results in a table (as shown below). Die score

2 2 Question 3. Observe your table results. Which score occurred most frequently? Answers will vary, but the of each die score will be close to 10. Given that only 60 trials are done, there may still be significant variation in the frequencies. Question 4. Compare your results with some other students near you. What do you notice? Answers will vary, but the of each die score will be close to 10. Given that only 60 trials are done, there may still be significant variation in the frequencies. To simulate the rolling of two dice, and adding the resulting numbers, we can add two random integers. Question 5. What command would simulate the sum of the two numbers when two dice are rolled? randint(1, 6) + randint(1, 6) Question 6. Try this command a number of times on your calculator. a) What is the smallest value you can obtain as a sum of the two random integers? 2 b) What is the largest value you can obtain as a sum of the two random integers? 12 Question 7. Now try the command 100 times, and record the results in a table (as shown below). Die sum Question 8. Comment on the distribution of results you obtained here. The die sum frequencies will vary, but the middle sums (e.g. 6, 7 and 8) should occur most frequently.

3 3 Part 2: Analysing results graphically Generating a table of results for an experiment such as this would be time consuming if we were to conduct a large number of trials, and so we will use the calculator to help us store, visualise and analyse the results using statistical features. To begin with, note that the command randint(1,6,600) diescore will generate 600 integers between 1 and 6, and store the results in a statistical variable named diescore. This can be used to generate and store the results of a large number of trials for further analysis. To do this on the TI-Nspire CAS Press MENU-5 to select Probability and then press 4 (Random) and then 2 (Integer). Type the following to complete the command randint(1,6,600) diescore (Note that the key for storing to a variable is CTRL-VAR.) The results of the 600 trials are displayed as a bracketed list as shown right. Question 9. How many times would you expect the number 1 to occur in these 600 trials? Explain your reasoning. We would expect a score of 1 to occur about 100 trials out of 600 (in the long term). It is difficult to scroll through this list of 600 numbers to count the of each number. However, the calculator has a counting feature that can provide this information. For instance, the command countif(diescore,1) will report the of the number 1 in the 600 stored trials. [Note: The countif command can be found by pressing the CATALOG key or just type it directly).] Question 10. Use this command to complete a table for the 600 trials. Die score Question 11. Compare these frequencies with your answer to question 9. Are these results what you expected? Explain your answer.

4 4 We can obtain a relative of each score by dividing the by the total number of trials. Question 12. Use this method to calculate the relative of obtaining a die score of 1. Use the formula that follows to assist. of scoring 1 = of scoring 1 Number of trials The experimental of scoring a 1 will be close to, but not necessarily exactly 100, so the relative will be close to 1/6 or Question 13. Copy and complete this table for your previous results. Die score (fraction) (decimal approx.) Question 14. What would you expect the relative of a die score of 1 to get close to if you conducted this trial a very large number of times? Express your answer as a fraction. The relative will be close to 1/6 or The results of the experiment can be viewed graphically. To do this, we will create a histogram of the results. On the TI-Npire CAS Press MENU-5 to add a Data and Statistics page. Click at the bottom of the screen to add the variable diescore to the plotting screen. (This creates a dot plot of the results of the 600 trials.) Press MENU-1 to select Plot Type and select 3 to select Histogram. The results of the 600 trials are displayed as a histogram. [Note that the columns are for numbers in the range 1 <2, 2 <3, and so on, rather than centred on the numbers 1 to 6.

5 5 Question 15. How would you expect this histogram to change in appearance if you were displaying the results of a much larger number of trials? It would be expected that the histogram columns become closer in height; that is, their frequencies will converge to same number as the number of trials increase. This is because each score on the die is equally likely. Now we will use this method to analyse the sum of two dice. Question 16. What command using the randint function will simulate the results of 1000 rolls of two dice, and store the results in a statistical variable named diesum? randint(1,6,1000) + randint(1,6,1000) diesum Press CTRL-LEFT ARROW to re-display the Calculator page, and enter this command. Question 17. Use the countif command to calculate the of a sum of 2 in the 1000 stored trials. Answers will vary, but the expected number would be 1/36 x 1000, or about 28. Question 18. Copy this table and use this command to complete the table for the 1000 trials. Die sum (fraction) (decimal approx.) Question 19. Are these relative frequencies what you expected? Explain your answer.

6 6 Question 20. Use your CAS to display a histogram of these results. Draw a sketch of this histogram and comment on its shape. Click on each column to verify the frequencies listed in the table above. Sample screen shown below Part 3: Long-term relative To work out what we expect to happen in the two-dice sum experiment in the long run, we will look at all the possible outcomes when two dice are rolled. Question 21. Copy and complete the following table to display all the possible outcomes when rolling two dice. Die 1 score ,1 2,1 3,1 4,1 5,1 6,1 2 1,2 2,2 3,2 4,2 5,2 6,2 Die 2 score 3 1,3 2,3 3,3 4,3 5,3 6,3 4 1,4 2,4 3,4 4,4 5,4 6,4 5 1,5 2,5 3,5 4,5 5,5 6,5 6 1,6 2,6 3,6 4,6 5,6 6,6 Question 22. For this experiment, how many possible outcomes are there in total? 36

7 7 The probability of a particular sum can be calculated as follows. Probability of a sum of 3 = Number of outcomes with a sum of 3 Total possible outcomes = Question 23. Use the above method to calculate the probability of obtaining a sum of 2, 3, , and then complete the table below. Sum No. of ways to obtain this sum Probability (fraction) Probability (decimal) /36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/ Question 24. Compare these theoretical probabilities with the relative frequencies obtained in your answer to question 17 and 18, and comment.

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