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## Summary:

- Realizations of very extreme outcomes from distributions with finite variance and infinite support.

## Main Question or Discussion Point

Distributions with finite variance and infinite support suggest non-zero, but negligible probability of very extreme outcomes. But how small is negligible, and how improbable is actually impossible?

For example, the adult male height in the US can be roughly characterized as N~(175, 9) in centimeters. [1] The tallest man within the last century was over 250cm, and the shortest male adult from the same period was only 55cm tall, the length of a large healthy newborn. The probability of exceeding each in their respective tail-ward direction, according to the distribution assumption, is

P(h>250) = 3e-19,

P(h<55) = 1e-37.

These numbers are in any conventional sense negligible, yet they both occurred within such a small span of human history, and other cases likely existed in the past. [2]

Even if we characterize the probability of at least one such outcome realizing in repeated individual trials based on the total number of men born during this period (about 6e9 since 1990, generously including a significant number of boys who did not survive into adulthood), we still have the very improbable

P(at least one > 250) = 2e-9

P(at least one <55) = 6e-28

If we model number of hurricanes in a year in the Atlantic as Poisson(6), then the comparable number of hurricanes to the probabilities in order of such infinitesimitude would be [3]:

P(25) = 4e-9

P(39) = 3e-19

P(49) = 5e-28

P(59) = 1e-37

The maximum number of hurricanes in the past 50 years was 15, at P(15)=9e-2, or 7 orders of magnitude than the chance of 25. At the same probability that a 250cm giant is born among us in a century, we would never expect 25 hurricanes in a year (at least not yet), let alone 40, 50, and 60.

So for hurricanes, one in a 250 million is impossible (without climate change, at least). For human beings, as numerous as we are, one in a billion is a reasonable expectation to occur, but one in a billion? One in a quintillion? Even one in a undecillion is apparently possible.

There are about 1e21 stars in the universe, so these probabilities are tiny even compared to the inverse of a universal scale. Maybe it can be put into the context of number of elementary particles in the universe, about 1e86.

But we would never expect a 5-meter tall man, not with 1e-86 probability, not even with 1e-1e100000 probability, both of which are allowed by such distributions.

So where is the line drawn?

(1) Granted it could be modeled more accurately as log-normal, but the generality of the discussion remains.

(2) Yes, the probability of any point outcome of a continuous distribution is zero yet they happen anyway, but let's not split this particular hair since any point outcome can be modeled as within a range.

(3) There's some small dispersion and the events are correlated, but the point is the same.

For example, the adult male height in the US can be roughly characterized as N~(175, 9) in centimeters. [1] The tallest man within the last century was over 250cm, and the shortest male adult from the same period was only 55cm tall, the length of a large healthy newborn. The probability of exceeding each in their respective tail-ward direction, according to the distribution assumption, is

P(h>250) = 3e-19,

P(h<55) = 1e-37.

These numbers are in any conventional sense negligible, yet they both occurred within such a small span of human history, and other cases likely existed in the past. [2]

Even if we characterize the probability of at least one such outcome realizing in repeated individual trials based on the total number of men born during this period (about 6e9 since 1990, generously including a significant number of boys who did not survive into adulthood), we still have the very improbable

P(at least one > 250) = 2e-9

P(at least one <55) = 6e-28

If we model number of hurricanes in a year in the Atlantic as Poisson(6), then the comparable number of hurricanes to the probabilities in order of such infinitesimitude would be [3]:

P(25) = 4e-9

P(39) = 3e-19

P(49) = 5e-28

P(59) = 1e-37

The maximum number of hurricanes in the past 50 years was 15, at P(15)=9e-2, or 7 orders of magnitude than the chance of 25. At the same probability that a 250cm giant is born among us in a century, we would never expect 25 hurricanes in a year (at least not yet), let alone 40, 50, and 60.

So for hurricanes, one in a 250 million is impossible (without climate change, at least). For human beings, as numerous as we are, one in a billion is a reasonable expectation to occur, but one in a billion? One in a quintillion? Even one in a undecillion is apparently possible.

There are about 1e21 stars in the universe, so these probabilities are tiny even compared to the inverse of a universal scale. Maybe it can be put into the context of number of elementary particles in the universe, about 1e86.

But we would never expect a 5-meter tall man, not with 1e-86 probability, not even with 1e-1e100000 probability, both of which are allowed by such distributions.

So where is the line drawn?

(1) Granted it could be modeled more accurately as log-normal, but the generality of the discussion remains.

(2) Yes, the probability of any point outcome of a continuous distribution is zero yet they happen anyway, but let's not split this particular hair since any point outcome can be modeled as within a range.

(3) There's some small dispersion and the events are correlated, but the point is the same.