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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.

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Probabilities are units used to quantify statements like "I don't need umbrella today, because it's unlikely to rain". They measure what they measure because this is how we defined them. What you mention are different possible interpretations of probability. There are different possible interpretations, because "probability does not exist" as said by Bruno de Finetti, it is just something we use to quantify statements like "likely", "unlikely", "equally possible" etc.

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  • $\begingroup$ OK, I see, in effect the definition of a "5% event" is something that happens 1 in 20 times, right? And probability theory lets us manipulate such quantities in a consistent manner. Makes sense, thanks. I'll read up the article - it starts a bit heavy on the meta, but maybe that's what I need! $\endgroup$ – Bennet Dec 13 '19 at 8:14
  • $\begingroup$ @Bennet not only "1 in 20", you mentioned yourself other interpretations of probability. $\endgroup$ – Tim Dec 13 '19 at 16:10
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By definition (or measurement), low probability events rarely happen.

But your example of your choice to take an umbrella suggests your underlying inquiry really has more to do with decision-making under uncertainty.

You might find Daniel Kahneman's book "Thinking, fast and slow" to be a good guide to understanding such issues.

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  • $\begingroup$ I only used the decision process as an example, as it directly anchors behaviours to reasonable expectations. $\endgroup$ – Bennet Dec 13 '19 at 8:15
  • $\begingroup$ As a person who thinks of himself as being logical, I used to think that behaviors (including various decisions I made) were based on reasonable expectations. Until I read Kahneman. $\endgroup$ – Doug Dame Dec 14 '19 at 3:30

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