Intriguing problems whose solutions seem to run counter to intuition are sometimes discussed theoretically in probability textbooks; under close scrutiny, they often appear to originate from an incomplete specification of the sampling procedures. Interest in these oddities is not purely academic; the world of real point processes (with its multitude of applications in science and engineering) exhibits its own ‘‘paradoxes’’ that follow from certain ambiguities in the random choice of an event. The paper discusses some probabilistic subtleties that can be easily illustrated by ordinary demonstration experiments in the physics laboratory.