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Discrete Random Variables: Lab II

Module by: Dr. Barbara Illowsky, Susan Dean

Summary: 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

Note:

For a detailed game description, refer here.

Note:

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 XX size 12{X} {} = number of matches. Theoretically, XX ~ ­­
  5. Let YY size 12{Y} {} = profit per game.

Organize the Data

In the chart below, fill in the YY size 12{Y} {} value that corresponds to each XX 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:
    • a. x ¯ = x ¯ = size 12{ {overline {x}} ={}} {}
    • b. s x = s x = size 12{s rSub { size 8{x} } ={}} {}
    • c. y ¯ = y ¯ = size 12{ {overline {y}} ={}} {}
    • d. s y = s y = size 12{s rSub { size 8{y} } ={}} {}
  3. Explain what x¯x¯ size 12{ {overline {x}} } {} represents.
  4. Explain what y¯y¯ size 12{ {overline {y}} } {} represents.
  5. Based upon the experiment:
    • a. What was the average profit per game?
    • b. Did this represent an average win or loss per game?
    • c. How do you know? Answer in complete sentences.
  6. Construct a histogram of the empirical data
    Figure 1
    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 XX size 12{X} {} and YY size 12{Y} {} based on the distribution from the section titled "The Procedure".

  1. X X size 12{X} {} Y Y size 12{Y} {} P X = x = P Y = y P X = x = P Y = y size 12{P left (X=x right )=P left (Y=y right )} {}
    0    
    1    
    2    
    3    
  2. Calculate the following
    • a. μ x = μ x = size 12{μ rSub { size 8{x} } ={}} {}
    • b. σ x = σ x = size 12{σ rSub { size 8{x} } ={}} {}
    • c. μ y = μ y = size 12{μ rSub { size 8{y} } ={}} {}
  3. Explain what μxμx size 12{μ rSub { size 8{x} } } {} represents.
  4. Explain what μyμy size 12{μ rSub { size 8{y} } } {} represents.
  5. Based upon theory:
    • a. What was the expected profit per game?
    • b. Did the expected profit represent an average win or loss per game?
    • c. How do you know? Answer in complete sentences.
  6. Construct a histogram of the theoretical distribution.
    Figure 2
    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):

Note:

RFRF = relative frequency

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

  1. P X = 3 = P X = 3 = size 12{P left (X=3 right )={}} {} ________________________
  2. P 0 < X < 3 = P 0 < X < 3 = size 12{P left (0<X<3 right )={}} {} ________________________
  3. P X ≥ 2 = 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 = RF X = 3 = size 12{ ital "RF" left (X=3 right )={}} {} ________________________
  2. RF 0 < X < 3 = RF 0 < X < 3 = size 12{ ital "RF" left (0<X<3 right )={}} {} ________________________
  3. RF X ≥ 2 = RF X ≥ 2 = size 12{ ital "RF" left (X >= 2 right )={}} {} ________________________

Discussion Question

  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. Does it appear that the data fit the distribution in (1)? In 1 - 3 complete sentences, explain why or why not.
  4. Suppose that the experiment had been repeated 500 times. Which chart (from (2) or (4)) would you expect to change? Why? How might the chart change?

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