To clarify, I would say that nature provides randomness, man provides probability that is a measure of randomness.

But speaking of probability, many probability density are asymmetrical so why randomness should imply symmetry ?

Or maybe I didn't understand the question.

Best

S.

This is only a preliminary introductory answer to clarify the question and admit that nature produces randomness/disorder which is an increase in entropy S according to the indisputable second law of thermodynamics dS>=0.

However, for example, an isolated system subject to Dirichlet boundary conditions, it can be shown that in the time-dependent 4D process,

an increase in entropy leads to more symmetry.

In other words, the probability of nature itself operates in 4D x-t unit space and produces a kind of symmetry.

From my point of view on this topic it sometimes seem to be symmetric but its not.

well, given a new (and very recent, btw) research finding

"that specific circular ribonucleic acids (RNAs) can stick to DNA in cells and cause mutations that result in cancer." - a cite from:

http://www.china.org.cn/world/Off_the_Wire/2023-06/09/content_86789032.htm

then how should this finding be compared with statement like "nature itself operates ... and produces a kind of symmetry" - what a symmetry is seen here when the result is cancer mutations - technically, resulting and coming from randomness in nature ...

** Answer II-Continued.

Below we offer three expressive photos of the majestic symmetry of nature.

Comments are left to Researchgate contributors.

https://apis.mail.yahoo.com/ws/v3/mailboxes/@.id==VjN-nqMHzvLPi99T7WYdPGruLiBPsNaM9325iLKvWyyUaNzDCCWgVBinIVTuF3JZIeYDOstHGQ2SvOzuBlPC52QM-g/messages/@.id==AHZGGhBLR-9HZISLnw7oaEu-mME/content/parts/@.id==2/thumbnail?appid=YMailNorrin&downloadWhenThumbnailFails=true&pid=2

Orange Symmetry

https://apis.mail.yahoo.com/ws/v3/mailboxes/@.id==VjN-nqMHzvLPi99T7WYdPGruLiBPsNaM9325iLKvWyyUaNzDCCWgVBinIVTuF3JZIeYDOstHGQ2SvOzuBlPC52QM-g/messages/@.id==AHZGGhBLR-9HZISLnw7oaEu-mME/content/parts/@.id==3/thumbnail?appid=YMailNorrin&downloadWhenThumbnailFails=true&pid=3

Earthquake symmetry

https://apis.mail.yahoo.com/ws/v3/mailboxes/@.id==VjN-nqMHzvLPi99T7WYdPGruLiBPsNaM9325iLKvWyyUaNzDCCWgVBinIVTuF3JZIeYDOstHGQ2SvOzuBlPC52QM-g/messages/@.id==AHZGGhBLR-9HZISLnw7oaEu-mME/content/parts/@.id==4/thumbnail?appid=YMailNorrin&downloadWhenThumbnailFails=true&pid=4

Butterfly symmetry

From mathematical point of view, your question has not any sense!

A question with sense can be: What is the probability that randomness produces symetry( in nature )? The answer is: it cannot be rigorously decided!

In my opinion, the symmetry created by nature is most closely described by a class of theorems, generically called the "central limit theorem". Deviations from the conditions of the central limit theorem (dependence of factors, dominance of one or more factors, etc.) generate asymmetry, which is observed in nature much more often than symmetry.