Discrete Distribution (Lucky Dice Experiment)
Class Time:
Names:
- The student will compare empirical data and a theoretical distribution to determine if a Tet gambling game fits a discrete distribution.
- The student will demonstrate an understanding of long-term probabilities.
- one “Lucky Dice” game or three regular dice
Procedure
Round answers to relative frequency and probability problems to four decimal places.
- The experimental procedure is to bet on one object. Then, roll three Lucky 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 object?
- Choose one object to place a bet on. Roll the three Lucky Dice. Count the number of matches.
- Let X = number of matches. Theoretically, X ~ B(______,______)
- Let Y = profit per game.
Organize the DataIn (Figure), 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.
x y Frequency Relative Frequency 0 1 2 3 - Calculate the following:
- = _______
- sx = ________
- = _______
- sy = _______
- Explain what represents.
- Explain what represents.
- Based upon the experiment:
- What was the average profit per game?
- Did this represent an average win or loss per game?
- How do you know? Answer in complete sentences.
- Construct a histogram of the empirical data.
Theoretical DistributionBuild the theoretical PDF chart for x and y based on the distribution from the Procedure section.
-
x y P(x) = P(y) 0 1 2 3 - 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 (Figure) or (Figure) to change, and how would it change? Why? Why wouldn’t the other table change?