[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Here is an attempt to get around the fact that we

cannot choose a “random integer.”

• What, intuitively, is the probability that a “randomly chosen” positive integer is a multiple of 3?
• Let [math]P_3(N)[/math] be the probability that an integer, chosen at random between 1 and [math]N[/math], is a multiple of 3 (since the sample space is finite, this is a legitimate probability). Show that the limit
  [[math]] P_3 = \lim_{N \to \infty} P_3(N) [[/math]]
  exists and equals 1/3. This formalizes the intuition in (a), and gives us a way to assign “probabilities” to certain events that are infinite subsets of the positive integers.
• If [math]A[/math] is any set of positive integers, let [math]A(N)[/math] mean the number of elements of [math]A[/math] which are less than or equal to [math]N[/math]. Then define the “probability” of [math]A[/math] as
  [[math]] P(A) = \lim_{N \to \infty} A(N)/N\ , [[/math]]
  provided this limit exists. Show that this definition would assign probability 0 to any finite set and probability 1 to the set of all positive integers. Thus, the probability of the set of all integers is not the sum of the probabilities of the individual integers in this set. This means that the definition of probability given here is not a completely satisfactory definition.
• Let [math]A[/math] be the set of all positive integers with an odd number of digits. Show that [math]P(A)[/math] does not exist. This shows that under the above definition of probability, not all sets have probabilities.