I've been trying to accept it, but it just can't seem to make sense to me. When we talk about the probability of a plane crashing, of being hit by a drunk driver, of being killed by a shark etc., we always refer to a probability that was measured ex post, namely as a ratio of (cases occurred)/(cases occurred + cases not occurred).
I know that it is one of the definitions of probability, but does it really represent a probability? After all, what we're actually doing is estimate the probability of an event happening in the future, based on how many times the event has happened in the past, which, per se, does not guarantee in any way that the event will keep happening with the same ratio. It would make sense if the state of the world kept repeating the same set of conditions over and over again (but here we're deviating from mathematics into philosophy and the Nietzschean concept of eternal recurrence), however I doubt this is the case.
Also, the aggregate measure of probability does not say much on probability at an individual level: if, for example, 1 out of 1 million planes crashes , that doesn't really mean that, if I take 1 million planes, then 1 of them will crash. Not even if I take 10 million planes will 10 crash.
I just can't wrap my head around it, but this concept of probability keeps being used in everyday life. Can anyone shed some light on it?
Posteriori probability is a Bayesian concept, not a Frequentist one. Under a Bayesian perspective, probability isn't a ratio of occurrences, it's a measure of belief. The posteriori probability is a measure of your belief given some prior beliefs (the prior distribution) and some information.