5.17 Discrete random variables: lab ii

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

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 $X$ = number of matches. Theoretically, $X~B(\_\_\_\_\_\_,\_\_\_\_\_\_)$
5. Let $Y$ = profit per game.

Organize the data

In the chart below, 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.

1. Complete the table.

   x	y	Frequency	Relative Frequency
   0
   1
   2
   3

2. Calculate the Following:
   • a $\overline{x}=$
   • b $s_x=$
   • c $\overline{y}=$
   • d $s_y=$
3. Explain what $\overline{x}$ represents.
4. Explain what $\overline{y}$ 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

   

Theoretical distribution

Build the theoretical PDF chart for $X$ and $Y$ based on the distribution from the section titled "The Procedure".

1. $x$	$y$	$P\left(x\right)=P\left(y\right)$
   0
   1
   2
   3

2. Calculate the following
   • a ${\mu }_x=$
   • b ${\sigma }_x=$
   • c ${\mu }_y=$
3. Explain what ${\mu }_x$ represents.
4. Explain what ${\mu }_y$ 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.

   

Use the data

Calculate the following (rounded to 4 decimal places):

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

1. $P\left(x=3\right)=$ $\text{\_\_\_\_\_\_\_\_\_\_\_\_}$
2. $\text{\_\_\_\_\_\_\_\_\_\_\_\_}$
3. $P\left(x\ge 2\right)=$ $\text{\_\_\_\_\_\_\_\_\_\_\_\_}$

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

1. $\text{RF}\left(x=3\right)=$ $\text{\_\_\_\_\_\_\_\_\_\_\_\_}$
2. $\text{\_\_\_\_\_\_\_\_\_\_\_\_}$
3. $\text{RF}\left(x\ge 2\right)=$ $\text{\_\_\_\_\_\_\_\_\_\_\_\_}$

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?