# 4.8: Discrete Distribution (Playing Card Experiment)

Name: ______________________________

Section: _____________________________

Student ID#:__________________________

*Work in groups on these problems. You should try to answer the questions without referring to your textbook. If you get stuck, try asking another group for help.*

**Student Learning Outcomes**

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

**Supplies**

- One full deck of playing cards

**Procedure**

The experimental procedure 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 ten times.
- Record the number of diamonds picked.
- Let \(X\) = number of diamonds. Theoretically, \(X\) ~
*B*(_____,_____)

**Organize the Data**

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

- Construct a histogram of the empirical data.
**Figure 4.8.1**

**Theoretical Distribution**

- 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:
- \(\mu\) = ____________
- \(\sigma\) = ____________

- Construct a histogram of the theoretical distribution.
**Figure 4.8.2**

**Using the Data**

Note 4.8.1

*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*(\(x \geq\) 8) = _______________________

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*(1 < \(x\) < 4) = _______________________*RF*(\(x \geq\) 8) = _______________________

**Discussion Questions**

For 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 and empirical distributions. Use complete sentences.
- Describe the three most significant differences between the graphs or distributions of the theoretical and empirical distributions.
- Using your answers from 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 orTable to change, and how would it change? Why? Why wouldn’t the other table change?