Question

Lets use Excel to simulate rolling two 8-sided dice and finding the rolled sum.

• Open a new Excel document.

• Click on cell A1, then click on the function icon fx and select Math&Trig, then select RANDBETWEEN.

• In the dialog box, enter 1 for bottom and enter 8 for top.

• After getting the random number in the first cell, click and hold down the mouse button to drag the lower right corner of this first cell, and pull it down the column until 25 cells are highlighted. When you release the mouse button, all 25 random numbers should be present.

• Repeat these four steps for the second column, starting in cell B1.

• Put the rolled sum of two dice in the third column: Highlight the first two cells in the first row and click on AutoSum icon. Once you receive the sum of two values in the third cell, drag the lower right corner of this cell, C1, down to C25. This will copy the formula for all 25 rows. We now have 25 trials of our experiment.

Once these steps are completed, attach a screenshot of your Excel file to your assignment.

(a) Find the theoretical probability that the rolled sum of both dice is 8.

(b) Based on the results of our experiment of 25 trials, obtain the relative frequency approximation to the probability found in (a). You can do so in Excel in two different ways: i) create the histogram of the third column data, then scroll the mouse over the relevant bar - this will give you the frequency with which you can determine the relative frequency; or ii) in a cell, type the function COUNTIF(C1:C25,8)

(c) Generate the frequency distribution histogram of your experiment of 25 trials, and copy it to a Word document. Make sure to add a title to your histogram.

(d) Repeat the simulation for 100 and 1000 trials, and calculate the relative frequency for each, and create the frequency distribution histogram - resize the 3 histograms so that all 3 fit beside each other in a row.

(e) Identify which of the 3 relative frequencies for ’8’ is the closest value to the theoretical probability found in (a). Briefly explain how these experiments demonstrate the Law of Large Numbers.

(f) Identify the shape of the probability distribution (uniform, bell-curved, right-skewed or left-skewed).

Answer 1

Ans.

=RANDBETWEEN(1,8)	=RANDBETWEEN(1,8)	=SUM(A1:B1)
=RANDBETWEEN(1,8)	=RANDBETWEEN(1,8)	=SUM(A2:B2)
=RANDBETWEEN(1,8)	=RANDBETWEEN(1,8)	=SUM(A3:B3)
=RANDBETWEEN(1,8)	=RANDBETWEEN(1,8)	=SUM(A4:B4)
=RANDBETWEEN(1,8)	=RANDBETWEEN(1,8)	=SUM(A5:B5)
=RANDBETWEEN(1,8)	=RANDBETWEEN(1,8)	=SUM(A6:B6)
=RANDBETWEEN(1,8)	=RANDBETWEEN(1,8)	=SUM(A7:B7)
=RANDBETWEEN(1,8)	=RANDBETWEEN(1,8)	=SUM(A8:B8)
=RANDBETWEEN(1,8)	=RANDBETWEEN(1,8)	=SUM(A9:B9)
=RANDBETWEEN(1,8)	=RANDBETWEEN(1,8)	=SUM(A10:B10)
=RANDBETWEEN(1,8)	=RANDBETWEEN(1,8)	=SUM(A11:B11)
=RANDBETWEEN(1,8)	=RANDBETWEEN(1,8)	=SUM(A12:B12)
=RANDBETWEEN(1,8)	=RANDBETWEEN(1,8)	=SUM(A13:B13)
=RANDBETWEEN(1,8)	=RANDBETWEEN(1,8)	=SUM(A14:B14)
=RANDBETWEEN(1,8)	=RANDBETWEEN(1,8)	=SUM(A15:B15)
=RANDBETWEEN(1,8)	=RANDBETWEEN(1,8)	=SUM(A16:B16)
=RANDBETWEEN(1,8)	=RANDBETWEEN(1,8)	=SUM(A17:B17)
=RANDBETWEEN(1,8)	=RANDBETWEEN(1,8)	=SUM(A18:B18)
=RANDBETWEEN(1,8)	=RANDBETWEEN(1,8)	=SUM(A19:B19)
=RANDBETWEEN(1,8)	=RANDBETWEEN(1,8)	=SUM(A20:B20)
=RANDBETWEEN(1,8)	=RANDBETWEEN(1,8)	=SUM(A21:B21)
=RANDBETWEEN(1,8)	=RANDBETWEEN(1,8)	=SUM(A22:B22)
=RANDBETWEEN(1,8)	=RANDBETWEEN(1,8)	=SUM(A23:B23)
=RANDBETWEEN(1,8)	=RANDBETWEEN(1,8)	=SUM(A24:B24)
=RANDBETWEEN(1,8)	=RANDBETWEEN(1,8)	=SUM(A25:B25)

3	3	6
2	4	6
1	2	3
1	3	4
3	7	10
8	6	14
2	5	7
6	6	12
5	5	10
4	5	9
3	5	8
7	8	15
4	4	8
4	7	11
5	5	10
7	8	15
6	8	14
5	5	10
2	5	7
7	6	13
7	6	13
2	8	10
1	1	2
8	4	12
8	5	13

=COUNTIF($C$1:$C$25,8)

2

8	1	9
3	6	9
3	7	10
7	1	8
3	1	4
4	4	8
8	4	12
5	6	11
2	7	9
4	6	10
1	5	6
5	8	13
8	8	16
1	8	9
2	1	3
2	7	9
7	1	8
2	4	6
8	7	15
8	8	16
4	8	12
3	1	4
4	8	12
1	2	3
3	2	5
2	2	4
3	8	11
4	3	7
6	1	7
6	5	11
5	5	10
1	1	2
5	8	13
6	7	13
5	1	6
8	3	11
2	5	7
6	4	10
2	5	7
4	2	6
4	2	6
1	4	5
4	4	8
8	2	10
4	5	9
2	1	3
3	4	7
6	5	11
2	1	3
3	1	4
1	2	3
1	7	8
5	3	8
6	7	13
8	3	11
7	2	9
7	6	13
3	3	6
3	1	4
7	8	15
1	4	5
5	3	8
2	4	6
5	2	7
2	3	5
5	5	10
2	8	10
1	1	2
6	3	9
1	8	9
7	3	10
1	2	3
5	4	9
2	3	5
8	6	14
5	5	10
8	2	10
8	2	10
5	2	7
3	7	10
7	3	10
4	6	10
6	8	14
5	6	11
8	2	10
2	1	3
8	3	11
7	8	15
6	7	13
5	1	6
5	1	6
5	1	6
5	4	9
2	4	6
3	3	6
1	3	4
3	3	6
7	4	11
5	2	7
8	4	12
	=countif(C1:C100,8)	7

118 counts

n	Frequency
25	2
100	7
1000	118

The shape of data is right-skewed .as n is increase the data follows parent distribution.