Problem:
A non-zero digit is chosen in such a way that the probability of choosing digit is . The probability that the digit is chosen is exactly the probability that the digit chosen is in the set
Answer Choices:
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E.
Solution:
Let be the probability that the digit chosen is one of . Note that
We seek a set of digits with probability . Thus
Note 1. In this solution, we have not used the fact that the logs are base . However, this fact is necessary for the problem to make sense; otherwise the union of all possibilities (i.e, picking some digit from to ) does not have probability .
Note 2. If one collects a lot of measurements from nature (say, the lengths of American rivers in miles), the fraction of the time that the first significant (i.e., nonzero) digit is is approximately . In particular, the distribution is not uniform, e.g., is the first digit about of the time. This counterintuitive fact, sometimes called Benford's Law, can be explained if one assumes that the distribution of natural constants is independent of our units of measurement.