Here are some things to think about:
 Can you find an interesting set S so that S_{n} is finite but Pr(S) doesn’t
exist? Can you find an interesting set S so that S_{n} is bounded but Pr(S)
doesn’t exist?
 What’s the probability that three integers chosen at random are relatively
prime? You can do a numerical simulation and even set up an infinite
product, but no one knows the exact values for the infinite product you’ll
create. It is known that the value is irrational.
 Learn about the Bernoulli numbers B_{n}. These are numbers that are defined by
the pattern
B_{0}  = 1  

B_{0} + 2B_{1}  = 0  

B_{0} + 3B_{1} + 3B_{2}  = 0  

B_{0} + 4B_{1} + 6B_{2} + 4B_{3}  = 0   
Yes, these numbers are from Pascal’s triangle. It turns out that
This generalizes the formula
to sums of reciprocals of even powers. Do you see how? Very little is known
about odd powers.
 What’s the probability that a prime number will leave a remainder of 3 when
you divide it by 4?