4.9: Discrete Distribution (Dice Experiment Using Three Regular Dice)
- Page ID
- 63331
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Class Time:
Names:
- The student will compare empirical data and a theoretical distribution to determine if a dice experiment game fits a discrete distribution.
- The student will demonstrate an understanding of long-term probabilities.
Supplies
- Three regular six-sided dice
Procedure
Round answers to relative frequency and probability problems to four decimal places.
- The experimental procedure is to bet on a specific number appearing on the dice. Then, roll three dice and count the number of matches. The number of matches will decide your profit.
- What is the theoretical probability of one die matching the the specific number?
- Choose one number to place a bet on. Roll the three dice. Count the number of matches.
- Let X = number of matches. Theoretically, X ~ B(______,______)
- Let Y = profit per game.
Organize the Data
In Table \(\PageIndex{1}\), fill in the y value that corresponds to each x value. Next, record the number of matches picked for your class. Then, calculate the relative frequency.
- Complete the table.
Table \(\PageIndex{1}\) x y Frequency Relative Frequency 0 1 2 3
2. Calculate the following:
a. \(\bar{x}=\) \(\_\_\_\_\)
b. \(s_x=\) \(\_\_\_\_\)
c. \(\bar{y}=\) \(\_\_\_\_\)
d. \(s_y=\) \(\_\_\_\_\)
3. Explain what \(\bar{x}\) represents.
4. Explain what \(\bar{y}\) represents.
5. Based upon the experiment:
a. What was the average profit per game?
b. Did this represent an average win or loss per game?
c. How do you know? Answer in complete sentences.
6. Construct a histogram of the empirical data.
Theoretical Distribution
Build the theoretical PDF chart for x and y based on the distribution from the Procedure section.
-
Table \(\PageIndex{2}\) x y P(x) = P(y) 0 1 2 3 Table 4.19
- Calculate the following:
- μx = _______
- σx = _______
- μx = _______
- Explain what μx represents.
- Explain what μy represents.
- Based upon theory:
- What was the expected profit per game?
- Did the expected profit represent an average win or loss per game?
- How do you know? Answer in complete sentences.
- Construct a histogram of the theoretical distribution.
Use the Data
RF = relative frequency
Use the data from the Theoretical Distribution section to calculate the following answers. Round your answers to four decimal places.
- P(x = 3) = _________________
- P(0 < x < 3) = _________________
- P(x ≥ 2) = _________________
Use the data from the Organize the Data section to calculate the following answers. Round your answers to four decimal places.
- RF(x = 3) = _________________
- RF(0 < x < 3) = _________________
- RF(x ≥ 2) = _________________
Discussion QuestionFor questions 1 and 2, consider the graphs, the probabilities, the relative frequencies, the means, and the standard deviations.
- Knowing that data vary, describe three similarities between the graphs and distributions of the theoretical and empirical distributions. Use complete sentences.
- Describe the three most significant differences between the graphs or distributions of the theoretical and empirical distributions.
- Thinking about your answers to questions 1 and 2, does it appear that the data fit the theoretical distribution? In complete sentences, explain why or why not.
- Suppose that the experiment had been repeated 500 times. Would you expect Table \(\PageIndex{1}\) or Table \(\PageIndex{2}\) to change, and how would it change? Why? Why wouldn’t the other table change?