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 positive probability of occurring?

Answer 1

Answer: if you have infinity points, which i will asume are the events, they cant have the same probability because then the probability will not be normalized, because in graph of prob vs variable, you will se infinite area under the curve if the probability is constant.

And yes, can all points have positive probability of occurring, but besides you medium value (the bell for example) you will see an asintotic decrease to the zero.