10.10: Collaborative Activity
- Page ID
- 64734
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)For this activity you will be supplied with three four-sided dice, two six-sided dice, and one twelve-sided die. This activity will consider how the frequency distributions of the sum of the four-sided dice, the sum of the six-sided dice, and the results of a throw of the twelve-sided die. To begin the activity, one member of the group should be designated as the dice roller and one member of the group should be designated as the data collector.
The four-sided dice will be used first. The dice roller will roll all three dice fifty times. For each roll the sum of the three dice will be observed, and the result will be recorded in a data table like the one shown in Table 10.20. As an example, consider fifty rolls of three four-sided die. Suppose that the first roll of the three four-sided dice that we observe a 2, 4, and 1. The sum of these is 7, and this is recorded in the data table as shown in Table 10.21. Suppose that the second roll of the three four-sided dice results in 4, 4, and 3. The sum is 11, and this is recorded in the data table as shown in Table 10.21. The remaining rolls are recorded in a similar manner. Example rolls are shown in Table 10.21.
Table 10.20 Data table for the collaborative activity.
|
Sum of Three Four-Sided Dice |
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|
Sum of Two Six-Sided Dice |
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|
One Twelve-Sided Die |
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Table 10.21 Data table for the collaborative activity.
|
Sum of Three Four-Sided Dice |
|||||||||
|
7 |
11 |
10 |
6 |
10 |
8 |
6 |
6 |
7 |
7 |
|
9 |
8 |
5 |
7 |
10 |
11 |
6 |
5 |
10 |
9 |
|
8 |
7 |
9 |
6 |
8 |
12 |
8 |
11 |
6 |
4 |
|
5 |
7 |
8 |
10 |
4 |
9 |
10 |
9 |
7 |
6 |
|
6 |
6 |
6 |
9 |
7 |
8 |
11 |
10 |
6 |
10 |
|
Sum of Two Six-Sided Dice |
|||||||||
|
6 |
3 |
5 |
5 |
9 |
3 |
6 |
7 |
6 |
12 |
|
11 |
8 |
8 |
7 |
5 |
5 |
7 |
9 |
7 |
11 |
|
5 |
7 |
8 |
6 |
8 |
8 |
7 |
6 |
3 |
7 |
|
11 |
9 |
8 |
8 |
11 |
8 |
7 |
10 |
7 |
4 |
|
10 |
5 |
5 |
6 |
8 |
6 |
6 |
11 |
5 |
11 |
|
One Twelve-Sided Die |
|||||||||
|
11 |
4 |
4 |
6 |
7 |
11 |
10 |
6 |
10 |
4 |
|
8 |
6 |
11 |
11 |
5 |
11 |
11 |
3 |
4 |
1 |
|
8 |
7 |
9 |
5 |
10 |
12 |
11 |
9 |
5 |
6 |
|
12 |
12 |
10 |
1 |
9 |
2 |
5 |
5 |
3 |
3 |
|
7 |
3 |
6 |
7 |
1 |
10 |
2 |
10 |
10 |
7 |
Next, the set of two six-sided dice are used. The dice roller will roll both dice fifty times. For each roll the sum of the two dice and each individual roll will be observed. The sum will be recorded in a data table like the one shown in Table 10.20, and the individual rolls will be recorded in a table like the one shown in Table 10.22. In this table the smaller of the rolls will be recorded first and the larger of the two rolls will be recorded second. As an example, consider fifty rolls of two six-sided dice. Suppose that the first roll of the two six-sided dice results in 2 and 4. The sum is 6, and this is recorded in the data table as shown in Table 10.21. In the second data table we will record 2 as the smaller of the two rolls and 4 as the larger of the two rolls, as shown in the example data recorded in Table 10.23. Suppose that the second roll of the two six-sided dice results in 1 and 2. The sum is 3, and this is recorded in the data table as shown in Table 10.21. Similarly, the smaller and larger numbers are recorded in Table 10.23. The remaining rolls are recorded in a similar manner. Example rolls are shown in Tables 10.21 and 10.23. Finally, the twelve-sided die is used. The dice roller will roll the die fifty times. For each roll, the outcome will be observed and the result will be recorded in a data (see Table 10.21).
Table 10.22 Data table for the collaborative activity.
|
Minimum of Two Six-Sided Dice |
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|
Maximum of Two Six-Sided Dice |
|||||||||
Table 10.23 Example data table for the collaborative activity.
|
Minimum of Two Six-Sided Dice |
|||||||||
|
1 |
1 |
1 |
2 |
4 |
1 |
1 |
3 |
1 |
6 |
|
5 |
4 |
3 |
3 |
2 |
1 |
3 |
3 |
2 |
5 |
|
2 |
1 |
4 |
1 |
3 |
2 |
3 |
1 |
1 |
2 |
|
5 |
3 |
2 |
2 |
5 |
3 |
1 |
4 |
3 |
1 |
|
5 |
2 |
2 |
2 |
2 |
3 |
2 |
5 |
2 |
5 |
|
Maximum of Two Six-Sided Dice |
|||||||||
|
5 |
2 |
4 |
3 |
5 |
2 |
5 |
4 |
5 |
6 |
|
6 |
4 |
5 |
4 |
3 |
4 |
4 |
6 |
5 |
6 |
|
3 |
6 |
4 |
5 |
5 |
6 |
4 |
5 |
2 |
5 |
|
6 |
6 |
6 |
6 |
6 |
5 |
6 |
6 |
4 |
3 |
|
5 |
3 |
3 |
4 |
6 |
3 |
4 |
6 |
3 |
6 |
Once the data have been collected, you and your group will need to construct four frequency distributions. The first three frequency distributions correspond to the sums of the three four-sided dice, the two six-sided dice, and the twelve-sided die. Each possible sum with correspond to a class for each of these frequency tables.
Considering the example data, we begin with the sums of three four-sided dice. When rolling three four-sided dice, the possible sums of the dice can be equal to 3, 4, 5, 6, 7, 8, 9, 10, 11, or 12. Therefore we construct a frequency distribution with these classes and count the number of times each sum was observed in our data. See Table 10.24. For the first entry in the table, we note that no sums added up to 3, so the frequency is 0, along with the relative frequency and the percentage. The next possible sum is 4, and we observe two instances where a sum of 4 was observed, so the frequency is 4, and the relative frequency is 4÷50=0.08 or 8% as indicated in Table 10.24. The remaining entries in Table 10.24 are computed in a similar manner. The frequency distributions for the sum of the two six-sided die and the twelve sided die are also constructed the same way, with the frequency distributions for the example data given in Tables 10.25 and 10.26.
Table 10.24 Example frequency distribution for the sum of three four-sided dice.
|
Sum |
Frequency |
Relative Frequency |
Percentage |
|
3 |
0 |
0.00 |
0% |
|
4 |
2 |
0.04 |
4% |
|
5 |
3 |
0.06 |
6% |
|
6 |
11 |
0.22 |
22% |
|
7 |
8 |
0.16 |
16% |
|
8 |
7 |
0.14 |
14% |
|
9 |
6 |
0.12 |
12% |
|
10 |
8 |
0.16 |
16% |
|
11 |
4 |
0.08 |
8% |
|
12 |
1 |
0.02 |
2% |
|
50 |
1.00 |
100% |
Table 10.25 Example frequency distribution for the sum of two six-sided dice.
|
Sum |
Frequency |
Relative Frequency |
Percentage |
|
2 |
0 |
0.00 |
0% |
|
3 |
3 |
0.06 |
6% |
|
4 |
1 |
0.02 |
2% |
|
5 |
8 |
0.16 |
16% |
|
6 |
8 |
0.16 |
16% |
|
7 |
9 |
0.18 |
18% |
|
8 |
9 |
0.18 |
18% |
|
9 |
3 |
0.06 |
6% |
|
10 |
2 |
0.04 |
4% |
|
11 |
6 |
0.12 |
12% |
|
12 |
1 |
0.02 |
2% |
|
50 |
1.00 |
100% |
Table 10.26 Example frequency distribution for the sum of two six-sided dice.
|
Sum |
Frequency |
Relative Frequency |
Percentage |
|
1 |
3 |
0.06 |
6% |
|
2 |
2 |
0.04 |
4% |
|
3 |
4 |
0.08 |
8% |
|
4 |
4 |
0.08 |
8% |
|
5 |
5 |
0.10 |
10% |
|
6 |
5 |
0.10 |
10% |
|
7 |
5 |
0.10 |
10% |
|
8 |
2 |
0.04 |
4% |
|
9 |
3 |
0.06 |
6% |
|
10 |
7 |
0.14 |
14% |
|
11 |
7 |
0.14 |
14% |
|
12 |
3 |
0.06 |
6% |
|
50 |
1.00 |
100% |
Finally, we will construct a cross-classified frequency table for the minimum and the maximum of the two six-sided die rolls reported in Table 10.23. To construct this table, note that the possible values for both the minimum and maximum of the two dice are 1, 2, 3, 4, 5, and 6. Therefore, construct a cross classified table like the one shown in Table 10.27 that shows the results for the example data. Once the blank table is constructed, you will need to count how often each possible cross-classified minimum and maximum occurs. Remember, these observations correspond to the same roll of the two dice, so you will need to match the entry in the top part of Table 10.23 with the corresponding entry in the bottom part of Table 10.23. For the example data, there were no instances where both the minimum and the maximum of the two dice were 1, and therefore this entry is 0, with a relative frequency of 0.00 and a percentage of 0%. The next entry corresponds to the case where the minimum of the two rolls was 1 and the maximum of the two rolls was 2. In the example data, there are three instances where this occurs, and hence the corresponding frequency recorded on the table is 3, with a relative frequency of 3÷50=0.06, or 6%. The remaining entries in the table are computed in a similar manner.
Table 10.27 Example cross classified frequency table for the minimum and maximum of the rolls of the two six-sided dice for the collaborative activity.
|
Maximum |
||||||||
|
Minimum |
1 |
2 |
3 |
4 |
5 |
6 |
Total |
|
|
1 |
0 0.00 0% |
3 0.06 6% |
1 0.02 2% |
2 0.04 4% |
5 0.10 10% |
2 0.04 4% |
13 0.26 26% |
|
|
2 |
0 0.00 0% |
0 0.00 0% |
6 0.12 12% |
2 0.04 4% |
2 0.04 4% |
4 0.08 8% |
14 0.28 28% |
|
|
3 |
0 0.00 0% |
0 0.00 0% |
1 0.02 2% |
5 0.10 10% |
3 0.06 6% |
2 0.04 4% |
11 0.22 22% |
|
|
4 |
0 0.00 0% |
0 0.00 0% |
0 0.00 0% |
2 0.04 4% |
1 0.02 2% |
1 0.02 2% |
4 0.08 8% |
|
|
5 |
0 0.00 0% |
0 0.00 0% |
0 0.00 0% |
0 0.00 0% |
1 0.02 2% |
6 0.12 12% |
7 0.14 14% |
|
|
6 |
0 0.00 0% |
0 0.00 0% |
0 0.00 0% |
0 0.00 0% |
0 0.00 0% |
1 0.02 2% |
1 0.02 2% |
|
|
Total |
0 0.00 0% |
3 0.06 6% |
8 0.16 16% |
11 0.22 22% |
12 0.24 24% |
16 0.32 32% |
50 1.00 100% |
|
Questions
- Compare the frequency tables for the rolls of three four-sided dice, two six-sided dice, and the twelve-sided die. What differences do you observe? Are there areas where more of the observations are concentrated, or are the frequencies evenly distributed throughout the classes?
- Consider the cross-classified frequency table for the minimum and the maximum of the rolls of the two six-sided dice. What patterns do you observe when you look at the cross-classified frequencies?
- Suppose that someone tells you that they had an observation of 6, and that they don’t remember what dice it came from. If you had to guess based on what you observed in your frequency distributions, would you conclude that the observation was from the three four-sided dice, the two six-sided dice, or the single twelve-sided die?
- Similarly, suppose that someone tells you that they had an observation of 3, and that they don’t remember what dice it came from. What would you guess based on your observed frequency distributions?
- Suppose that someone tells you that they had an observation of 1, and they forget what dice they rolled. Would you conclude that the observation was from the three four-sided dice, the two six-sided dice, or the single twelve-sided die?