A thought today from Pchem:
Infinity is a relative term. One meter away from the nucleus of an atom is infinity, and 10 billion billion kilometers away from the sun is infinity. Since infinity is a general concept, rather than a number, it can be defined anywhere. So, if we consider the probability of finding an electron such-and-such a distance from the nucleus, we can find the probability that it will be from that point inwards, or the probability of finding the electron between two points by doing the same method but subtracting the smaller value. We know that the probability of finding the electron converges to 0 at infinity, but infinity can be anywhere we set it to be. Supposing you want to find the probability of finding the electron on Mars (as was the example given today), you can find the probability between "Nucleus and Mars" (A very high number), and you can then find the probability between"Just beyond Mars and Infinity". Then you can subtract "Nucleus and Mars" probability from "Just beyond Mars and Infinity" to get a real probability of finding the electron on Mars. I think this all arises because we can set infinity anywhere we want (which is necessary for the concept of infinity to be of any use).