### A Puzzle: Two-step Selection of a Digit

Here is a simple problem in probability.

(1) Pick a number k between 1 and 9. Assume all digits are equally likely.

(2) Pick a number m in the range from 1 to k.

What is the probability distribution for the number m?

A graph of the probability distribution is shown in the figure here.

Probability distribution for a decimal digit selected in a two-step process.

Can you derive a formula for this probability distribution?

Can you generalise it to the range from 1 to 10^n?

Can you relate this problem to Benford’s Law [described here]?

Solution, and more on Benford’s Law, next week.