Discrete Distribution (Playing Card Experiment)
Class Time:
Names:
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 compare technology-generated simulation and a theoretical distribution.
- The student will demonstrate an understanding of long-term probabilities.
Supplies
- One full deck of playing cards
- One programming calculator
ProcedureThe 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 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 with playing cards in [link]. Then calculate the relative frequency.
| x | Frequency | Relative Frequency |
|———-
| 0 | \_\_\_\_\_\_\_\_\_\_ | \_\_\_\_\_\_\_\_\_\_ |
| 1 | \_\_\_\_\_\_\_\_\_\_ | \_\_\_\_\_\_\_\_\_\_ |
| 2 | \_\_\_\_\_\_\_\_\_\_ | \_\_\_\_\_\_\_\_\_\_ |
| 3 | \_\_\_\_\_\_\_\_\_\_ | \_\_\_\_\_\_\_\_\_\_ |
| 4 | \_\_\_\_\_\_\_\_\_\_ | \_\_\_\_\_\_\_\_\_\_ |
| 5 | \_\_\_\_\_\_\_\_\_\_ | \_\_\_\_\_\_\_\_\_\_ |
| 6 | \_\_\_\_\_\_\_\_\_\_ | \_\_\_\_\_\_\_\_\_\_ |
| 7 | \_\_\_\_\_\_\_\_\_\_ | \_\_\_\_\_\_\_\_\_\_ |
| 8 | \_\_\_\_\_\_\_\_\_\_ | \_\_\_\_\_\_\_\_\_\_ |
| 9 | \_\_\_\_\_\_\_\_\_\_ | \_\_\_\_\_\_\_\_\_\_ |
| 10 | \_\_\_\_\_\_\_\_\_\_ | \_\_\_\_\_\_\_\_\_\_ |
- Calculate the following:
-
= \_\_\_\_\_\_\_\_
- s = \_\_\_\_\_\_\_\_
- Construct a histogram of the empirical data.
![This is a blank graph template. The x-axis is labeled Number of diamonds. The y-axis is labeled Relative frequency.](/statistics-book/resources/fig-ch04_17_01.png)
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:
- μ = \_\_\_\_\_\_\_\_\_\_\_\_
- σ = \_\_\_\_\_\_\_\_\_\_\_\_
- Construct a histogram of the theoretical distribution.
![This is a blank graph template. The x-axis is labeled Number of diamonds. The y-axis is labeled Probability.](/statistics-book/resources/fig-ch04_17_02.png)
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 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 [link] or [link] to change, and how would it change? Why? Why wouldn’t the other table(s) change?