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.Round relative frequencies and probabilities to four decimal places.

The ProcedureThe 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. What is the theoretical probability of 1 die matching the object? Choose one object to place a bet on. Roll the 3 Lucky Dice. Count the number of matches. Let X size 12{X} {} = number of matches. Theoretically, X ~ 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. Complete the table. X Y Frequency Relative Frequency 0 1 2 3

Calculate the Following: a x ¯ = size 12{ {overline {x}} ={}} {} b s x = size 12{s rSub { size 8{x} } ={}} {} c y ¯ = size 12{ {overline {y}} ={}} {} d s y = size 12{s rSub { size 8{y} } ={}} {} Explain what x¯ size 12{ {overline {x}} } {} represents. Explain what y¯ size 12{ {overline {y}} } {} represents. Based upon the experiment: aWhat was the average profit per game? bDid this represent an average win or loss per game? cHow do you know? Answer in complete sentences. Construct a histogram of the empirical data

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". X size 12{X} {} Y size 12{Y} {} P X = x = P Y = y size 12{P left (X=x right )=P left (Y=y right )} {} 0 1 2 3

Calculate the following a μ x = size 12{μ rSub { size 8{x} } ={}} {} b σ x = size 12{σ rSub { size 8{x} } ={}} {} c μ y = size 12{μ rSub { size 8{y} } ={}} {} Explain what μx size 12{μ rSub { size 8{x} } } {} represents. Explain what μy size 12{μ rSub { size 8{y} } } {} represents. Based upon theory: aWhat was the expected profit per game? bDid the expected profit represent an average win or loss per game? cHow do you know? Answer in complete sentences. Construct a histogram of the theoretical distribution.

Use the Data Calculate the following (rounded to 4 decimal places): RF = relative frequency Use the data from the section titled "Theoretical Distribution" here: P X = 3 = size 12{P left (X=3 right )={}} {} ____________ P 0 < X < 3 = size 12{P left (0<X<3 right )={}} {} ____________ P X ≥ 2 = size 12{P left (X >= 2 right )={}} {} ____________ Use the data from the section titled "Organize the Data" here: RF X = 3 = size 12{ ital "RF" left (X=3 right )={}} {} ____________ RF 0 < X < 3 = size 12{ ital "RF" left (0<X<3 right )={}} {} ____________ RF X ≥ 2 = size 12{ ital "RF" left (X >= 2 right )={}} {} ____________

Discussion Question 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.) 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.) Does it appear that the data fit the distribution in (1)? In 1 - 3 complete sentences, explain why or why not. 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?