Question

1. Let’s use Excel to simulate rolling two 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 6 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 probability that the rolled sum of both dice is 5.

(b) Based on the results of our experiment of 25 trials, obtain the relative frequency approximation to the probability found in (a).

(c) Repeat the simulation for 50 and 100 trials, and calculate the relative frequency approximation to the probability in (a) for each. Which approximation has the closest value to the probability?

(d) Briefly explain how these experiments demonstrate the Law of Large Numbers.

Answer 1

Screenshot of the Excel file with A= outcome of DIce1 and B= outcome of Dice2 and C= sum of A and B

a) Total number of outcomes = 36

Sum is 5 when the dice1 and dice2 values are (1,4),(2,3),(3,2), (4,1)

Probabitty = 4/36 = 1/9.

b) Based on the result of our experiment of 25 trials-

number of times sum is 5 is 2.

relative frequency approximation = 2/25

relative frequnecy approximation to probablity found in (a) = 2/25 divide by 1/9 = 18/25

c) For 50 trials -

number of times sum is 5 is 5

relative frequency approximation = 5/50 = 1/10

relative frequnecy approximation to probablity found in (a) = 1/10 divide by 1/9 = 9/10

For 100 trials-

number of times sum is 5 is 11

relative frequency approximation = 11/100

relative frequnecy approximation to probablity found in (a) = 11/100 divide by 1/9 = 99/100

d) Law of Large numbers is that the results obtained from large number of trials should be close to expected value, and will tend to come closer to expected value as more trials are performed.

Our experiment proves the above theorem. As we simulations with more trials are performed, the relative frequency approximation comes closer to expected probability.