#### Discrete Distribution (Playing Card Experiment)

- The student will compare empirical data and a theoretical distribution to determine if an everyday experiment fits a discrete distribution.
- The student will compare technology-generated simulation and a theoretical distribution.
- The student will demonstrate an understanding of long-term probabilities.

- One full deck of playing cards
- Programmable calculator

Procedure for Empirical DataThe experimental procedure for empirical data is to pick one card from a deck of shuffled cards.

- The theoretical probability of picking a diamond from a deck is ________.
- Shuffle a deck of cards.
- Pick one card from it.
- Record whether it was a diamond or not a diamond.
- Put the card back and reshuffle.
- Do this a total of 10 times.
- Record the number of diamonds picked.
- Let
*X*= number of diamonds. Theoretically,*X*~*B*(_____,_____)

Procedure for SimulationRepeat the experimental procedure using a programmable calculator.

- Use the randInt function to generate data. Consider 1 to be spades, 2 to be hearts, 3 to be diamonds, and 4 to be clubs. Generate 10 draws of cards with four suits with randInt(1,4,10).
- Let $X=\text{numberofdiamonds}$. Theoretically,
*X*~*B*(_____,_____).

- Record the number of diamonds picked for your class with playing cards in Table 4.15. Then calculate the relative frequency.
*x*Frequency Relative Frequency 0 __________ __________ 1 __________ __________ 2 __________ __________ 3 __________ __________ 4 __________ __________ 5 __________ __________ 6 __________ __________ 7 __________ __________ 8 __________ __________ 9 __________ __________ 10 __________ __________ - Calculate the following:
- $\overline{x}$ = ________
*s*= ________

- Construct a histogram of the empirical data.

- Use Table 4.16 to record the number of diamonds picked for your class using the calculator simulation. Calculate the relative frequency.
*X*Frequency Relative Frequency 0 __________ __________ 1 __________ __________ 2 __________ __________ 3 __________ __________ 4 __________ __________ 5 __________ __________ 6 __________ __________ 7 __________ __________ 8 __________ __________ 9 __________ __________ 10 __________ __________ - Calculate the following:
- $\overline{x}$ = ________
*s*= ________

- Construct a histogram of the simulation data.

- Build the theoretical PDF chart based on the distribution in the Procedure section.
*x**P*(*x*)0 1 2 3 4 5 6 7 8 9 10 - Calculate the following:
*μ*= ____________*σ*= ____________

- Construct a histogram of the theoretical distribution.

Using the Data

#### NOTE

*RF* = relative frequency

Use the table from the Theoretical Distribution section to calculate the following answers. Round your answers to four decimal places.

*P*(*x*= 3) = ________*P*(1 <*x*< 4) = ________*P*(*x*≥ 8) = ________

Use the data from the Organize the Empirical Data section to calculate the following answers. Round your answers to four decimal places.

*RF*(*x*= 3) = ________*RF*(1 <*x*< 4) = ________*RF*(*x*≥ 8) = ________

Use the data from the Organize the Simulation Data section to calculate the following answers. Round your answers to four decimal places.

*RF*(*x*= 3) = ________*RF*(1 <*x*< 4) = ________*RF*(*x*≥ 8) = ________

Discussion QuestionsFor Questions 1 and 2, think about the shapes of the two 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, empirical, and simulation distributions. Use complete sentences.
- Describe the three most significant differences between the graphs or distributions of the theoretical, empirical, and simulation distributions.
- Using your answers from Questions 1 and 2, does it appear that the two sets of 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 4.15, Table 4.16, or Table 4.17 to change, and how would it change? Why? Why wouldn’t the other table(s) change?