Dice Activity

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Histograms - A 'Dice-y' Topic

Activity Guide

Part 1: Six-sided dice

(1) Is there a way to accurately predict the sum of the faces when two regular six-sided dice are rolled simultaneously (e.g., If a red die shows a 5 and a green die shows a 2, then the sum would be 7)? Why or why not?

(2) Predict the sum that will most frequently occur when a pair of dice is rolled ten times. Explain the criteria you used to make your prediction. Test your prediction by rolling the dice ten times. As you roll the dice, record the results in Table 1.

Table 1 Results of ten-trial test

Trial	Result - Red	Result - Green	Sum
1
2
3
4
5
6
7
8
9
10

(3) Use Table 2 to create a histogram of the resulting sums. What patterns, if any, do you notice in the histogram? If you performed an eleventh trial of the experiment, what sum would most likely be the outcome? How did you use your histogram to help form your prediction?

Table 2. Ten-trial histogram

Sum

2 3 4 5 6 7 8 9 10 11 12

(4a) Now simulate 10 trials of the dice-rolling experiment using your graphing calculator. Use the LIST features of your calculator to store the results of the10 trials.

How to simulate rolls of a die using the TI-73 graphing calculator:

This simulation uses the random integer generator capabilities of your calculator. The random integer generator requires three parameters for this activity: randInt (a,b,c) where 'a' is the lower bound of the random integers to be generated, 'b' is the upper bound, and 'c' is the number of random integers to be generated (e.g., randInt(1,6,10) will generate 10 random integers between 1 and 6 inclusive).

Be sure the list locations (L1, L2, and L3) that will be used to store the data for this simulation are empty before proceeding to Step 1. A quick way to clear ALL list entries is to press 2ND, MEM (the 0 key), 6 (ClrAllLists), ENTER.

Follow the steps below to simulate the rolling of the red die 10 times:

Step 1. Select LIST. Using the arrow keys, highlight the list with the column heading, L1 (see Screen 1).

Step 2. Select MATH, PRB, 2 (randInt), ENTER (see Screen 2).

Step 3. Type in randInt(1,6,10), ENTER (see Screen 3). This places the results from the red die into list L1 (see Screen 4). You may wish to save this information in a named list.

Screen 1 Screen 2 Screen 3 Screen 4

Step 4. Repeat Steps 1 through 3 for the green die, saving the results in a different list, L2.

To generate the sum of the two dice in L3, follow the steps below:

Step 5. Press LIST. Using the arrow keys, highlight the column heading, L3 (see Screen 5).

Step 6. Define L3 as L1+ L2 by pressing, 2ND, LIST (STAT), 1 (L1), ENTER, +, 2ND, LIST (STAT), 2 (L2), ENTER (see Screen 6). The entry by entry sums of L1 and L2 should now be in L3 (see Screen 7).

Screen 5 Screen 6 Screen 7

(4b) Using your graphing calculator, create a histogram of the results (sums) of the 10-trial experiment.

How to create a histogram using the TI-73 graphing calculator:

Before creating the histogram, be sure the Stat Plots are turned off. To turn off the Stat Plots, go to the home screen and press CLEAR, 2ND, Y= (Plot), 4 (PlotsOff), ENTER, ENTER.

The steps below will create a histogram from the data in L3:

Step 1. Press 2ND, Y= (Plots), 1 (Plot 1), ENTER. This displays the Plot 1 set-up screen (see Screen 8).

Screen 8

Step 2. Using the arrow keys, move the cursor to ON and press ENTER.

Step 3. Move the cursor to the histogram icon and press ENTER.

Step 4. Move the cursor to the Xlist row. Xlist needs to read 'L3', so press 2ND, LIST (STAT), 3 (List 3), ENTER (see Screen 9).

Screen 9

By default, the Freq row should read '1' because we are tabulating the frequencies in L3 only once.

Step 5. Press GRAPH. The graph may or may not appear at this point. Don't worry - yet.

Step 6. Press ZOOM, 7 (ZoomStat). A histogram should now appear (similar to that shown in Screen 10).

Screen 10

Step 7. Press WINDOW. This screen defines the viewing window. As a class, determine appropriate numbers for Xmin, Xmax, Ymin, Ymax, and Yscl. Generally it is best to have a negative Ymin value. This creates a space at the bottom of the screen where coordinates can be displayed without overwriting the graph. Press GRAPH once a uniform viewing window is set-up.

(4c) Investigate your histogram by pressing TRACE and manipulating the arrows keys. What are some trends that you notice in the data? What is the range of the data? What is mode of the sums? Find the relative frequency for each possible outcome. Compare and contrast the data in this histogram with the histogram you created in Task 3.

(5) Compare your histogram to others in the class. Compare relative frequencies. What, if any, overall trends do you notice?

(6) Predict what a histogram would look like based on the results of 50 rolls of the dice. Create a histogram based on your predictions.

(7a) Using your graphing calculator, simulate 50 trials of the dice-rolling experiment and create a histogram of the outcomes.

Tips for creating this histogram:

1. Clear out the old data in L1 (red), L2 (green), and L3 (sum). Your new data will be entered into these lists (to conserve memory).

2. The parameters for the random integer generator will now be randInt(1,6,50).

3. As a class, decide on an appropriate viewing window for this histogram.

(7b) Discuss the characteristics of this histogram in terms of axis labels and units, range and domain, and data trends. Find the relative frequency for each possible outcome. Which outcome had the greatest relative frequency? Compare this histogram to the histogram you created in Task 6. Compare your histograms to others in the class.

(7c) Is your calculator-generated 10-trial or 50-trial histogram more likely to look like your neighbor’s calculator-generated 10-trial or 50-trial histogram? Why do you think this is the case?

(8a) Repeat Tasks 6, 7a, 7b, and 7c for 400 trials.

Tips for creating this histogram:

1. Clear out the old data in L1 (red), L2 (green), and L3 (sum). Your new data will be entered into these lists (to conserve memory). If a memory error occurs, delete some unneeded programs/applications (check with your teacher beforehand) or reduce the number of trials from 400 to 300.

2. The parameters for the random integer generator will now be randInt(1,6,400).

3. As a class, decide on an appropriate viewing window for this histogram.

(8b) Make conjectures about how the number of trials conducted in this dice-rolling experiment affects the class-wide uniformity of the resulting histograms.

(9) If you conducted one more trial of this experiment, what would be the most likely outcome? Why? Is this prediction the same as the one you made in Task 1? Why or why not?

(10) Compare the relative frequencies of the sums that are less than 7 with the relative frequencies of the sums that are greater than 7. What, if any, sums have similar relative frequencies? Make conjectures about why these similarities might have occurred.

Part 2: Eight-sided dice

(11) Predict the sum that will most frequently occur when a pair of eight-sided dice is rolled ten times. Explain the criteria you used to make your prediction. Test your prediction by rolling the dice ten times.

(12) Predict what a histogram would look like based on the resulting sums of 400 rolls of a pair of eight-sided dice. Create a histogram based on your predictions.

(13a) Using your graphing calculator, simulate 400 trials of the eight-sided dice-rolling experiment and create a histogram of the outcomes.

(13b) Discuss the characteristics of this histogram in terms of axis labels and units, range and domain, and data trends. Find the relative frequency for each possible outcome. Which outcome had the greatest relative frequency? How does this histogram compare to the histogram you constructed in Task 12?

(14) Based on this histogram, which outcome is most likely to occur? Why? Is your answer the same as the one you made in Task 11? If not, why not?

(15) Compare and contrast this histogram to the histogram you created in Task 8. What similarities exist between the shapes of the two histograms?

Part 3: Twelve-sided dice

(16) Repeat Task 11 through Task 14 using a pair of twelve-sided dice.

(17) Compare and contrast the calculator-generated histogram in Task 16 to the histograms you created in Task 8 and Task 13. What similarities exist between the shapes of the three histograms? Where does the greatest relative frequency occur in the three histograms?

Part 4: Generalizing the results

(18) Fill in Table 3, assuming that N is an even number. How did you determine the sum with the greatest relative frequency in a 400 trial experiment for an N-sided pair of dice?

Table 3. Greatest relative frequency table

Sides on each die	Sum with greatest relative frequency in a 400 trial experiment
6
8
12
N

(19) Predict what a histogram would look like based on the results of 400 rolls of a pair of N-sided dice. Explain the criteria you used to make your prediction. Create a histogram, in as much detail as possible, based on your predictions.

(20) How would the histogram you created in Task 19 change if N were an odd number?

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Last modified on April 3, 2002.