Question

On a fair 6-sided die each number has an equal probability p of being rolled. When a fair die is rolled n times, the most likely outcome (the mean) is that each number will be rolled np times, with a standard deviation of σ=√(np(1-p)). Brandon rolls a die 200 times. He will conclude that the die is loaded (unfair) if the number of time any number is rolled is outside of the 1.5 standard deviations of the mean. What are the minimum and maximum number of times a number can be rolled without Brandon concluding that the die is loaded?

Answer 1

A 1.5 standard deviation above the mean is
np + 1.5 √(np(1-p))

We are given with
np = 200
p = 200/n
The standard deviation is
√200(1-200/n))

Substituting
200 + 1.5 √200(1-200/n))
By inspection, if the value of n is 200, then the radical will result to the value of 1. Only by increasing the value of n greater than 200 will the radical result to a value of less than 1 and decreasing the spread of the mean.
The answer is
the minimum is 201
the maximum is infinity