Is there a non-tautological explanation as to why low-probability events happen rarely? For example, if I know there is only 5% probability of rain, what is the exact logic that makes it OK to leave my umbrella at home?
Some reasons I considered, and rejected, include:
- Deviation from expectation: in expectation, over say 100 trials, there is an expectation of 5 successes (assuming p=0.05 as above), a small number, and over 100 realisations I'm OK getting wet 5 times. But expectation is only useful, because there is a low probability of getting a result far from it.
- Strong law of large numbers: almost surely, over an infinite number of realisations, the fraction of low probability events is small (5% in this case). But over a finite sample, we are really reduced to the above case. Again, low-probability events don't happen
- Expectation-based ones: maybe I have a utility function that penalises both carrying an umbrella, and getting wet, and my utility is maximised by leaving the umbrella behind. But that relies on the assumption that over the long run, my average utility will be close to the expectation - again, because average being far from expectation has a low probability.
So what is the link? We use expected-value- and expected-utility-based logic all the time in life, with good results, so there must be a logical link.