Q. 2.2 - A First Course in Probability

Question

Consider an experiment whose sample space consists of a countably infinite number of points. Show that not all points can be equally likely. Can all points have a positive probability of occurring?

Step-by-Step Solution

Answer

It is impossible for all points to be equally probable. It is impossible for all points to have a positive probability.

Step 1 Given Information.

Consider an experiment whose sample space consists of a countably infinite number of points.

Step 2 Explanation.

Let us consider the hypothesis that all points are equally likely.

Let $n$be the number of points and $p$the nonzero probability of each point.

Hence, $np=1.$ However, as $n$is infinite, so must be $np,$which therefore cannot equal$1$. It is thus impossible that all points have equal probability.

To guarantee that $np=1$it is necessary that $n$be finite. In other words, there must be a finite number of points with a positive probability of occurring, excluding from the sample space all other points, the probability of which being zero. It is impossible for an infinitely large number of points to have a positive probability of occurring.