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# Discrete Random Variables: Lab I

Summary: This module allows students to explore concepts related to discrete random variables through the use of a simple playing card experiment. Students will compare empirical data to a theoretical distribution to determine if the experiment fist a discrete distribution. This lab involves the concept of long-term probabilities.

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Class Time:

Names:

## Student Learning Outcomes:

• The student will compare empirical data and a theoretical distribution to determine if everyday experiment fits a discrete distribution.
• The student will demonstrate an understanding of long-term probabilities.

## Supplies:

• One full deck of playing cards

## Procedure

The experiment procedure is to pick one card from a deck of shuffled cards.

1. The theorectical probability of picking a diamond from a deck is:
2. Shuffle a deck of cards.
3. Pick one card from it.
4. Record whether it was a diamon or not a diamond.
5. Put the card back and reshuffle.
6. Do this a total of 10 times
7. Record the number of diamonds picked.
8. Let X=X= number of diamonds. Theoretically, XX ~

## Organize the Data

1. Record the number of diamonds picked for your class in the chart below. Then calculate the relative frequency.
Table 1
X Frequency Relative Frequency
0 __________ __________
1 __________ __________
2 __________ __________
3 __________ __________
4 __________ __________
5 __________ __________
6 __________ __________
7 __________ __________
8 __________ __________
9 __________ __________
10 __________ __________
2. Calculate the following:
• a. x¯ x =
• b. ss =
3. Construct a histogram of the empirical data.

## Theorectical Distribution

1. Build the theoretical PDF chart for X based on the distribution in the section above.
 X X size 12{X} {} P X = x P X = x size 12{P left (X=x right )} {} 0 1 2 3 4 5 6 7 8 9 10
2. Calculate the following:
• a. μ=μ= size 12{μ={}} {}________________________
• b. σ=σ= size 12{σ={}} {}________________________
3. Constuct a histogram of the theoretical distribution.

## Using the Data

Calculate the following, rounding to 4 decimal places:

### Note:

RF = relative frequency

Use the table from the section titled "Using the Data" here:

• P ( X = 3 = P(X=3=
• P ( 1 < X < 4 ) = P(1<X<4)=
• P ( X 8 = P(X8=

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

• RF ( X = 3 ) = RF(X=3)=
• RF ( 1 < X < 4 ) = RF(1<X<4)=
• RF ( X 8 ) = RF(X8)=

## Discussion Questions

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 Part I? In 1 - 3 complete sentences, explain why or why not.
4. Suppose that the experiment had been repeated 500 times. Which chart (from Part II or Part III) would you expect to change? Why? Why wouldn’t the other chart change? How might the chart change?

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