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The throwing of a one or two is called a success. The six dice are thrown together 64 times and the frequencies of the throws with 0, 1, 2,..., 6 successes are summed over all six pairs are as follows:

Success: 0 1 2 3 4 5 6
Frequency: 0 4 19 15 17 7 2

(a)What would be the theoretical probability of success and the mean and variance of the above frequency distribution is all the dice were fair?

My friend and I got different answers. I multiplied by the successes from 0,..6, and my friend multiplied by the frequencies.

Who is correct? enter image description here enter image description here

(c) Test the hypothesis that the mean and variance have their theoretical mean?

What kind of test would you suggest? 2-sample binomial test and we can approx with Z? And what kind of test for variance? Chi-sqaured?

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You are definitely correct to multiply the number of successes by their probabilities for E(X).

The result (2) should be intuitive since since you have a 1/3 chance on any given die roll and 6 dice. However, if you do this 64 times, you should multiply your results by 64 as well (i.e. 128 is the expected successes for 64 sets of 6 dice - 384 total rolls).

Here is a link on confidence intervals for binomial distributions - there are several options: http://en.wikipedia.org/wiki/Binomial_distribution#Confidence_intervals

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