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What makes an event rare? Waking up tomorrow in the morning seems to be a common event. Having a genetic disease that only occurs in 0.003% of the population may be considered rare.

Yet the first event can be described in a different way that makes it seem more unique: waking up tomorrow in the morning at your particular place, say New York, on your particular bed at the particular time of 7:47:08 AM. Now, the same event seems more rare.

Is rarity, then, an illusion? If so, why do we categorize some events as rare and others not?

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  • Armand is right. In 2020 I was diagnosed with a rare form of cancer. How rare? In the entire USA, there are only about 100 cases of it per year i.e., 1 case per 3.4 million people. The important thing here is the per 3.4 million, coupled with the fact that we are not talking about every person in that class discovering that they had Extramammary Paget's Disease in the same year, on the same day, at the same exact moment, on exactly the same body part.
    – niels nielsen
    Jun 5 at 4:27
  • *armand scored a goal there! Hip Hip Hooray! Other heads concur! Learnt some math from the question. Danke!
    – Agent Smith
    Jun 5 at 4:53
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    Your question shows that whether we categorize an event, such as waking up in the morning, as rare or not depends on how we describe it. So rarity depends on description, which we choose. It just means it depends on the context in which we place it. So it is a relative property (like, e.g. tall), not an illusion.
    – Ludwig V
    Jun 5 at 14:14
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    Every time you ask "is X an illusion", the answer is gonna be no.
    – Davor
    Jun 6 at 18:13
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    @RonJohn hmm, I can see why you would think that. My intended point was that 1, 2, 3, 4, 5, 6, 7 "seems" more rare in the sense of the question but actually has an equal probability to some arbitrary set of numbers like 11, 14, 17, 18, 21, 25, 31. So basically exactly what you said.
    – N. Virgo
    Jun 12 at 13:20

4 Answers 4

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Each event happens only once. You waking up on the morning of June 5th 2023 will ever happen only once.

Considering this, it makes no sense to speak about the rarity of one event. What you are really asking about is the rarity of a class, or category, of events. "Waking up in the morning" or "waking up in New York" are not events, but classes of events. And obviously the rarity of a class of events depends on how the class is defined. This definition will always be somewhat arbitrary, but is usually done with its relevance in regard to a certain goal in mind.

For example if I were to decide the priority of research effort for treating genetic diseases, it's relevant to compare which has the most occurrences (among other factors). If I were living in New York and trying to assess if I am partying too much, "waking up in my bed in New York" compared to "waking up in a stranger's bed in New Jersey" may be relevant categories of events to consider.

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I think you are confusing uniqueness with rarity. Every grain of sand on a beach if different from every other, but grains of sand are not rare. We use the term rare to refer to instances of objects or events whose distinctive attributes are of stand-out importance to us. The differences between one grain of sand and another are of no consequence to us, so we lump all the grains together as simply sand. A guitar played by Jimi Hendrix would, among hundreds of thousands of unique guitars, be counted as rare because we attach importance to one of its unique characteristics. If the guitar had never been played by Jimi, the remaining factors contributing to its uniqueness would me much less important, so we might not consider it rare. Rarity is no more an illusion than any other subjective quality we attribute to objects and events, such as beauty, usefulness, interest, worth and so on.

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You're forgetting to include context. Simple Bayesian inference. The "given" part of the probability. The chance that you wake up in New York, GIVEN ___ (where you live, etc), vs the chance you get a disease, GIVEN you were born ___ (as a human being, etc).

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Is rarity, then, an illusion?

All statements related to probability/stochastic only make sense in the context of a mathematical theory with firm, consistent definitions. Tell me your definitions, and I'll tell you the answer.

In no case is this related to our perception of it - as long as you work in a given (and stated) mathematical framework, there is no "illusion", at least not in a more concrete sense that any high level maths tends to look like magic to mere mortals.

Even if you don't state a proper probability framework, each human has some intuitive understanding of it, which still is one of several frameworks - just unstated.

A secondary and possibly overly nitpicky aspect is that if you call something an illusion, you maybe mean that "it looks like it exists, but it does not really". Now we are in the realm of existence, which is a very deep area of philosophy for sure. In any case, to give a quick answer, I'd say "rarity" is a concept, or in the best case an attribute of something - but it is not something which "exists" in the first place at all; therefore it cannot be an illusion. (And also, if you talk about rarity of events, the same holds: an event is not an object or something that exists, so its rarity can hardly be an illusion at all).

So the only sense in which rarity can be an illusion is if we're talking fully intuitively, and then it simply means that our definitions do not match - person A talks a different "probability language" than person B, so the concept of rarity for person A might just not make sense to person B, but in truth it's a translation issue more than a philosophical one. And certainly not a deep philosophical, existential issue.

If so, why do we categorize some events as rare and others not?

Because the framework we chose to work in dictates that it be so (or not).

Note that the brain, for all of its wondrous capabilities is really not good with probabilities; especially with very small ones, and especially when combined with very large or very small "costs" of individual events. Intuitively, I would expect any intuitive statement about probability to be wrong in more cases than not. This is why industries like insurance or casinos thrive as well as they do.

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