Question

A die is rolled 100 times. The total number of spots is 368 instead of the expected 350. Can this be explained as a chance variation, or is the die loaded?

A die is rolled 1000 times. The total number of spots is 3680 instead of the expected 3500. Can this be explained as a chance variation, or is the die loaded?

Answer 1

When we roll a single die then possible outcomes are 1, 2, 3,4 ,5 and 6.

Since each of the six outcomes are equally likely so probability of getting each outcome is 1/6.

The average is:

Now,

The variance is

The standard deviation is

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Let T shows the total number of spots in 100 rolls. According to central limit theorem, sampling distribution of sample sum will be approximately normal distribution with following parameters  

and standard deviation

The z-score for T = 368-0.5 is

The z-score for T = 368+0.5 is

The probability of getting 368 dots s

Since this probability is less than 0.05 so it seems that die is loaded.

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Let T shows the total number of spots in 1000 rolls. According to central limit theorem, sampling distribution of sample sum will be approximately normal distribution with following parameters  

and standard deviation

The z-score for T = 3680-0.5 is

The z-score for T = 3680+0.5 is

The probability of getting 368 dots s

Since this probability is less than 0.05 so it seems that die is loaded.