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?
A 1.5 standard deviation above the mean is np + 1.5 <span>√(np(1-p))
We are given with np = 200 p = 200/n The standard deviation is </span><span>√200(1-200/n))
Substituting 200 + 1.5 </span>√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
