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Consider the interval [0, 1]. Assume you randomly pick a real number in that interval with...

Consider the interval [0, 1]. Assume you randomly pick a real number in that interval with a probability density that is constant on that interval and write the number in decimal code. Give an argument for the following statement by using suitable Theorems and ideas from class: With probability equal to one the relative number of the figures “9” you see amongst the first n digits approaches 1/10 as n tends to infinity.

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Answer #1

The first n digits are a random sample of the digits 0,1,2,3,4,5,6,7,8,9 such that each of the n places filled with any of the 10 digits with equal probability. Let f be the number of "9"s in the first n places. The relative number of "9"s is f/n. By the strong law of large numbers the sample mean f/n converges to the population mean 1/10 as n tends to infinity.

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