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This module allows students to apply concepts related to discrete distributions to a simple dice experiment. Students will compare empirical data and a theoretical distribution to determine if the game fits a discrete distribution. This experiment involves the concept of long-term probabilities.

Class Time:

Names:

Student learning outcomes:

  • 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.

Supplies:

  • 1 game “Lucky Dice” or 3 regular dice
For a detailed game description, refer here . (The link goes to the beginning of Discrete Random Variables Homework. Please refer to Problem #14.)
Round relative frequencies and probabilities to four decimal places.

The procedure

  1. The experiment procedure is to bet on one object. Then, roll 3 Lucky Dice and count the number of matches. The number of matches will decide your profit.
  2. What is the theoretical probability of 1 die matching the object? _________
  3. Choose one object to place a bet on. Roll the 3 Lucky Dice. Count the number of matches.
  4. Let X size 12{X} {} = number of matches. Theoretically, X ~ B (______,______)
  5. Let Y size 12{Y} {} = profit per game.

Organize the data

In the chart below, fill in the Y size 12{Y} {} value that corresponds to each X size 12{X} {} value. Next, record the number of matches picked for your class. Then, calculate the relative frequency.

  1. Complete the table.
    x y Frequency Relative Frequency
    0
    1
    2
    3
  2. Calculate the Following:
    • x ¯ = size 12{ {overline {x}} ={}} {}
    • s x = size 12{s rSub { size 8{x} } ={}} {}
    • y ¯ = size 12{ {overline {y}} ={}} {}
    • s y = size 12{s rSub { size 8{y} } ={}} {}
  3. Explain what x ¯ size 12{ {overline {x}} } {} represents.
  4. Explain what y ¯ size 12{ {overline {y}} } {} represents.
  5. 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.
  6. Construct a histogram of the empirical data
    Blank graph with relative frequency on the vertical axis and number of matches on the horizontal axis.

Theoretical distribution

Build the theoretical PDF chart for X size 12{X} {} and Y size 12{Y} {} based on the distribution from the section titled "The Procedure".

  1. x size 12{x} {} y size 12{y} {} P x = P y size 12{P left (X=x right )=P left (Y=y right )} {}
    0
    1
    2
    3
  2. Calculate the following
    • μ x = size 12{μ rSub { size 8{x} } ={}} {}
    • σ x = size 12{σ rSub { size 8{x} } ={}} {}
    • μ y = size 12{μ rSub { size 8{y} } ={}} {}
  3. Explain what μ x size 12{μ rSub { size 8{x} } } {} represents.
  4. Explain what μ y size 12{μ rSub { size 8{y} } } {} represents.
  5. 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.
  6. Construct a histogram of the theoretical distribution.
    Blank graph with relative frequency on the vertical axis and number of matches on the horizontal axis.

Use the data

Calculate the following (rounded to 4 decimal places):

RF = relative frequency

Use the data from the section titled "Theoretical Distribution" here:

  1. P x = 3 = size 12{P left (x=3 right )={}} {} ____________
  2. P 0 < x < 3 = size 12{P left (0<x<3 right )={}} {} ____________
  3. P x 2 = size 12{P left (x>= 2 right )={}} {} ____________

Use the data from the section titled "Organize the Data" here:

  1. RF x = 3 = size 12{ ital "RF" left (x=3 right )={}} {} ____________
  2. RF 0 < x < 3 = size 12{ ital "RF" left (0<x<3 right )={}} {} ____________
  3. RF x 2 = size 12{ ital "RF" left (x>= 2 right )={}} {} ____________

Discussion question

For questions 1. and 2., consider the graphs, the probabilities and relative frequencies, the means and the standard deviations.

  1. Knowing that data vary, describe three similarities between the graphs and distributions of the theoretical and empirical distributions. Use complete sentences. (Note: these answers may vary and still be correct.)
  2. Describe the three most significant differences between the graphs or distributions of the theoretical and empirical distributions. (Note: these answers may vary and still be correct.)
  3. Thinking about your answers to 1. and 2.,does it appear that the data fit the theoretical distribution? In 1 - 3 complete sentences, explain why or why not.
  4. Suppose that the experiment had been repeated 500 times. Which table (from "Organize the Data" or "Theoretical Distribution") would you expect to change? Why? How might the table change?

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Source:  OpenStax, Collaborative statistics. OpenStax CNX. Jul 03, 2012 Download for free at http://cnx.org/content/col10522/1.40
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