This is undefined for P = 1. If you claim that that function is a real-valued bijection between probabilities and odds then P = 1 doesn’t work so you’re begging the question. Always take care to not divide by zero.
Whether or not real-world events can have a probability of 0 or 1 is a different question than “are 0 and 1 probabilities?”. They most certainly are.
I agree with this one. Without probabilities of 0 and 1, it’s not merely that some proofs of theorems need to be revised, it’s that probability theory simply doesn’t work anymore, as its very axioms fall apart.
I can give a statement that is absolutely certain, e.g. “x is true given that x is true”. It doesn’t teach me much about real life experiences, but it is infinitely certain. Likewise with probability 0. Please note that the probability is assigned to the territory here, not the map.
The fact that I can’t encounter these probabilities in real life has to do with my limits of sampling reality and interpreting it, being a flimsy brain, rather than the limits of probability theory.
You may not want to believe that probability theory contains 0 and 1, but like many other cases, Math doesn’t care about your beliefs.
This is undefined for P = 1. If you claim that that function is a real-valued bijection between probabilities and odds then P = 1 doesn’t work so you’re begging the question. Always take care to not divide by zero.
Whether or not real-world events can have a probability of 0 or 1 is a different question than “are 0 and 1 probabilities?”. They most certainly are.
I agree with this one. Without probabilities of 0 and 1, it’s not merely that some proofs of theorems need to be revised, it’s that probability theory simply doesn’t work anymore, as its very axioms fall apart.
I can give a statement that is absolutely certain, e.g. “x is true given that x is true”. It doesn’t teach me much about real life experiences, but it is infinitely certain. Likewise with probability 0. Please note that the probability is assigned to the territory here, not the map.
The fact that I can’t encounter these probabilities in real life has to do with my limits of sampling reality and interpreting it, being a flimsy brain, rather than the limits of probability theory.
You may not want to believe that probability theory contains 0 and 1, but like many other cases, Math doesn’t care about your beliefs.