Question

You have 3 dice. Dice A is a fair dice while Dice B and Dice C are not fair. The odd outcomes of Dice B are twice as likely as the even outcomes while the even outcomes of Dice C are twice as likely as the odd outcomes. Each dice is thrown 5,000 times independently and you will use Excel to simulate this experiment.

1. Plot the histograms of the outcome from each dice and also the sum of the outcomes from the 3 dice. Comment on the shape of the histograms. Briefly comment on the histograms for the 3 dice individually - whether they are consistent with the theoretical results. For the sum of the outcomes from 3 dice (sum = 3,4,5,...,18), describe the distribution of the sum.
2. Obtain the (empirical) probability distribution of the outcomes of the three dice in the form of a table. You can use Excel to construct this table.
3. A member in your group repeats this experiment using three fair dice. He concludes that the result in 1.2 is identical to the experiment using Dice A, B and C. Do you agree with him? Give reasons as why. You can either obtain the theoretical values of the probabilities or empirical probabilities (recommended) of the outcomes from throwing the 3 fair dice 5000 times. Then compare the probability distributions to see whether they look like each other.

Answer 1

• You have 3 dice. Dice A is a fair dice while Dice B and Dice C are not fair. The odd outcomes of Dice B are twice as likely as the even outcomes while the even outcomes of Dice C are twice as likely as the odd outcomes. Each dice is thrown 5,000 times independently and you will use Excel to simulate this experiment. 1.1 Plot the histograms of the outcome from each dice and also the sum of the outcomes from the 3 dice. Comment on the shape of the histograms. 1.2 Obtain the (empirical) probability distribution of the outcomes of the three dice in the form of a table. You can use Excel to construct this table. 1.3 A member in your group repeats this experiment using three fair dice. He concludes that the result in 1.2 is identical to the experiment using Dice A, B and C. Here the probabilities are according to the given information.

• DATA -> DATA ANALYSIS -> HISTOGRAM

  Then select the input range as the probabilities provided and bin range as per your choice. Then click on cumulative distribution and frequency dialog box and click ok.

  For empirical distribution, again click on data and then data analysis and choose descriptive statistics and then provide the input range which will be again the probabilities provided. click the dialog box with summary statistics and click ok.

  Repeat the steps as done in first part with different probabilities (probabilites as dice A )AND THEN REPEAT THE STEPS FROM SECOND PART and you will see that the empirical distribution output comes out to be same.

  Column1
  Mean	0.166666667
  Standard Error	6.73172E-18
  Median	0.166666667
  Mode	0.166666667
  Standard Deviation	2.85603E-17
  Sample Variance	8.15688E-34
  Kurtosis	-2.266666667
  Skewness	1.09330348
  Range	0
  Minimum	0.166666667
  Maximum	0.166666667
  Sum	3
  Count	18