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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?

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  • $\begingroup$ You might be interested in the problem of induction. $\endgroup$
    – joriki
    Jun 7, 2016 at 17:37
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    $\begingroup$ You are wrong that probability equals the ratio (cases occurred) / (cases occured + cases not occurred). That's called frequency, and it need not be equal to probability. $\endgroup$
    – Lee Mosher
    Jun 7, 2016 at 17:48
  • $\begingroup$ I think that when you say that identifying the frequency of a certain kind of event with its "probability" is actually a kind of estimation of the probability. However I would not understimate that concept. In fact, I can say that the probability of guessing a number at the roulette is 1/37, following an apriori reasoning (assuming the fairness of the machine!). Hoewver, if I would instead look at the statitics of the casinos, I will arrive, for some misteroius reason, at a very very very similar conclusion. The meaning of the prob. from an empirical point of view is a matter of debate. $\endgroup$
    – guestDiego
    Jun 7, 2016 at 17:50
  • $\begingroup$ Of course, probability and frequency are distinct, but they're often used interchangeably when an a priori estimate is not possible, because as guestDiego said, they can lead to the same conclusion. However, spinning a roulette or rolling a die represent a rather simple system, while the probabilities (frequencies) that I'm referring to draw inferences on an (almost?) infinitely complex system (ultimately, the interaction of human and natural behavior), so I don't see why, just because inferences of this kind hold for simple systems, they should hold for complex ones. $\endgroup$
    – contenrico
    Jun 7, 2016 at 18:05
  • $\begingroup$ The problem of induction cited by joriki is, indeed, very relevant, but unfortunately does not provide a definite answer (like most philosophy, for that matter). $\endgroup$
    – contenrico
    Jun 7, 2016 at 18:09

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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.

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  • $\begingroup$ My bad, by "a posteriori probability" I don't mean the posterior probability in Bayesian statistics, but the empirical probability as a result of real-life measurement a posteriori. Apparently, "a posteriori probability" can mean both - en.wikipedia.org/wiki/A_posteriori_probability $\endgroup$
    – contenrico
    Jun 7, 2016 at 18:24

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