Minitab – Probability Distributions

 

Rolling Dice

This lab will involve examining probability distributions and expected values for when one fair die and two fair dice are rolled.

Roll One Fair Die

Let y = the number we see when one fair die is rolled.

Setting Up the Minitab Worksheet

Set up your Minitab worksheet to look similar to the table below.

C1

C2

y

P(y)

1

0.166667

2

0.166667

3

0.166667

4

0.166667

5

0.166667

6

0.166667

Simulating 1000 Rolls of One Die

    1. Label C3 "Roll y".
    2. Take one die and roll it 1000 times. Or let Minitab simulate the 1000 rolls.
    3. Choose Calc>Random Data>Discrete
    4. Generate 1000 rows of data
    5. Store in column: Roll y
    6. Values in: y
    7. Probabilities in: P(y)
    8. OK

Tally Results

    1. Choose Stat>Tables>Tally
    2. Variables: Roll y
    3. Select the Counts and Percents check boxes
    4. OK

Calculate the Mean, Standard Deviation, and Variance

    1. Choose Calc>Column Statistics
    2. Find the Mean of the Roll y column
    3. OK
    4. Repeat process except find the Standard Deviation of the Roll y column
    5. By hand (with a calculator) square the standard deviation to get the variance. Type it in the session window.

Rolling One Die 1000 More Times

We want to again roll a fair die 1000 times and find the mean, standard deviation, and variance for our new sample. Let z = the number we see when we roll a fair die.

Simulating 1000 Rolls of One Die

    1. Label C4 "Roll z".
    2. Take one die and roll it 1000 times. Or let Minitab simulate the 1000 rolls.
    3. Choose Calc>Random Data>Discrete
    4. Generate 1000 rows of data
    5. Store in column: Roll z
    6. Values in: y
    7. Probabilities in: P(y)
    8. OK

Tally Results

    1. Choose Stat>Tables>Tally
    2. Variables: Roll z
    3. Select the Counts and Percents check boxes
    4. OK

Calculate the Mean, Standard Deviation, and Variance

    1. Choose Calc>Column Statistics
    2. Find the Mean of the Roll z column
    3. OK
    4. Repeat process except find the Standard Deviation of the Roll z column
    5. By hand (with a calculator) square the standard deviation to get the variance. Type it in the session window.

Roll Two Fair Dice

Let x = the sum of the numbers we see when two fair dice are rolled. Therefore, x can be any number from 2 to 12. Fill out the rest of the table below by adding together the results of each row and column.

 

1

2

3

4

5

6

1

2

 

 

 

 

 

2

 

 

 

 

 

 

3

 

 

 

 

 

 

4

 

 

 

 

 

 

5

 

 

 

 

 

 

6

 

 

 

 

 

12

Setting Up the Minitab Worksheet

Set up the next four columns of your worksheet to look like those below.

C5

C6

C7

C8

x

Freq(x)

P(x)

xP(x)

 

 

 

 

Entering Values for C5

The x column should contain all the possible outcomes for a roll of two fair dice. Therefore, C5 should contain the numbers 2 through 12.

Entering Values for C6

Count the number of time each value of x appears in the table of sums you made for the rolls of two dice.

Calculating C7

    1. Choose Calc>Calculator
    2. Store results in variable: P(x)
    3. Expression: Freq(x)/SUM(Freq(x))
    4. OK

Find the Expected Value for x

    1. Choose Calc>Calculator
    2. Store results in variable: xP(x)
    3. Expression: x*P(x)
    4. OK
    5. Choose Calc>Column Statistics
    6. Statistic: Sum
    7. Input Variable: yP(y)
    8. OK

Simulating 1000 Rolls of Two Dice

    1. Label C9 "Roll x".
    2. Take two dice and roll them 1000 times. Or let Minitab simulate the 1000 rolls.
    3. Choose Calc>Random Data>Discrete
    4. Generate 1000 rows of data
    5. Store in column: Roll x
    6. Values in: x
    7. Probabilities in: P(x)
    8. OK

Tally Results

    1. Choose Stat>Tables>Tally
    2. Variables: Roll x
    3. Select the Counts and Percents check boxes
    4. OK

Calculate the Mean, Standard Deviation, and Variance

    1. Choose Calc>Column Statistics
    2. Find the Mean of the Roll x column
    3. OK
    4. Repeat process except find the Standard Deviation of the Roll x column
    5. By hand (with a calculator) square the standard deviation to get the variance. Type it in the session window.

Relationship Between the Means

What do you notice about the means for your sample rolls of one die and two dice? Explain your results. Type your answer into the session window.

Relationship Between the Variances

How is the variance for the roll of one die related to the variance for the roll of two dice? Why would it be difficult to see a relationship between the standard deviations? Explain your results. Type your answer into the session window.

Show in Theory

On a separate piece of paper, show in theory (an argument that would prove any example) the relationship between the means and variances for rolls of one die and two dice.

To Hand In:

Hand in your assignment in this order:
    1. Session window with Name at the top.
    2. History Window
    3. Theory explanation

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