**eklhad@ihnet.UUCP (K. A. Dahlke)** (05/27/85)

< 383722163088362149883726161744636203 > Last week, my meandering mind came across a problem I heard several years ago. The question put to me then: given a random positive integer, what is the probability that its decimal expansion contains a '9'. After a few minutes of thought, I proudly announced "1". Being a young, untrained high school student, I was quite confident, Especially since those around me agreed with my answer. Looking back, I realize the problem is not well defined. What does it mean to select an integer at random. When the set of choices is finite, a standard interpretation is assumed. Each of N choices is selected with a probability of 1/N. This becomes meaningless when the set is infinite. This problem requires another model. Adopting one interpretation, my answer was correct. What is the limit, as N approaches infinity, of the probability that a randomly selected integer from 1 to N contains a '9'. If N = 10^M, the probability is 1 - 0.9^M, which approaches 1. Other variants actually select an integer from the infinite set. Suppose we select integer I, with probability 1/2^I. In other words, a geometric progression: 1 with probability 1/2, 2 with probability 1/4, 3 with probability 1/8, etc. This one is not as easy to solve. What is the probability that an integer so selected contains a '9'. Feel free to try other density functions as well. Mail or post any thoughts. -- Karl Dahlke ihnp4!ihnet!eklhad