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.
in reduced form (
and 
coprime), a Ford circle is a circle with center at 
and radius 
. There is a Ford circle associated with every rational number. Every Ford circle is tangent to the horizontal axis and each two Ford circles are either tangent or disjoint from each other.
. Another example is the wave equation derived by Scottish mathematical physicist James Clerk Maxwell. Our modern world would not exist without the knowledge encapsulated in Maxwell’s equations. 
ThatsMaths 