How does one understand the connection between symmetry and randomness? In this famous book Physics from symmetry the author Jakob Schwichtenberg tells us all the existing physics can be derived by the symmetry of nature.In my opinion, the intuitive thing I can recall about symmetry is the dice or the coin, which has six or two probabilities respectively and you can derive the same probabilities($1/6$, or $1/2$) just from the symmetry. 

So my question is: are there any more profound connections between symmetry and randomness in physics?

 A: This may reasonably considered to be a list or an opinion/discussion question, it seems that way to me, and so off topic.
But rather than involving randomness, which I think makes things unnecessarily complicated, you could create a division between symmetry and non symmetry.
There are the obvious external symmetries of a snowflake, but there are also internal symmetries such as isospin, in which protons and neutrons are treated in the same way by the Strong Nuclear Force.
I have not read the book you mention, but (to me) the most profound result of symmetry is the Standard Model, upon which our current understanding of reality, (whatever that really is) is based.    
A: When it is mentioned the main role symmetries play in physics one does not mean symmetries of different configurations or arrangements such as the symmetries of dices, coins, snowflakes etc. It is the symmetries of the physical laws by themselves that are so important and this is the context of the book you cited.  In this sense, randomness is not in general fundamentally connected to symmetry, although it is for some systems such as ferromagnetic materials. 
A: This is a profound question. Firstly,  without a doubt, there must be some source of asymmetry for there to be a universe. How does it arise?
Ignoring the question of matter /anti-matter for a minute to simplify the thought experiment.
Take an imaginary new universe full of hot plasma. If that was totally evenly distributed; and, if the variables acting upon all points in space-time were balanced exactly; and, if there were no changes or movements in any fields at all... then there would have been nothing to cause the cooling plasma to gravitationally attract into disequil clumps of hydrogen gas to form galaxies.
However, we know that galaxies did form and we know gravity does exist and we know that quantum fields are constantly changing but in a random way: virtual particles popping in and out of existence all the time in a concept called quantum foam.
Now the question might be: if the change in the quantum foam is absolutely random (not just Hellishly complex and pseudorandom) then how would any gravitational attraction arise?
Take an 'ideal' circle: the curve is constant. If you zoom in infinitely, there would be no discrete pixels in our ideal circle, only a continuous curve. If you now take an imaginary quantum computer with staggering processing speed,  RAM etc, and give it an infinite amount of time to calculate PI, you could look for patterns in the calculated number to try and predict the next digits. As the ideal circle has complete symmetry, there are no disbalances and thus no predictability. The chance of 1-9 arising would always be the same. This absolute randomness arises out of absolute geometeric symmetry.
Now imagine you're a new hydrogen atom lying in a absolutely evenly distributed 3d-cloud of other hydrogen atoms. You're all pulling upon one another gravitationally to exactly the same extent. No movement is happening.
However,  you're living in a space-time that has quantum fields where random change occurs due to the appearance and disappearance of virtual particles. Imagine that no virtual particles have yet popped into existence---symmetry pervades still! Suddenly, a virtual particles pops into existence and this temporarily increases gravity minutely in one region of space, such that, the other hydrogen atoms locally around that region are attracted. This tiny movement towards that centre of attraction leads to more gravitational attraction and the tiny effect amplifies. This is what occurs in self-organising systems ie, systems forming out of the sudden and random appearence of an attractor. A galaxy starts to form rapidly.
It's not am isolated event though: across an infinite space-time of hydrogen gas, random attractors pop into existence frequently, leading to a roughly even distribution of galaxies.
Within a small volume of space-time (locally) there is an asymmetry of hydrogen: as more gas is pulled into the gravitational centre, less hydrogen resides in the periphery.
However, over a large enough volume, you get a clear fractal pattern of galaxies connected by filament like structures.

So randomness occurs precisely because---within any system of variables that exist within a dynamic state---the chance of the change in level of any variable is exactly the same as the chance of a change in level of any other variable. Such a fluid state seems to pervade quantum fields throughout space-time.
Putting it another way: if there were an asymmetry acting on quantum fields to produce fluctuations, such a system would likely exhibit predictable patterns. In such a universe, at a large scale, there wouldn't be an even distribution of galaxies.
There is another system which produces random noise and non-random forms: holograms.

When a reference lazer beam crosses another reflected lazer beam ---ie, that has reflected off of an object---the pattern that is generated upon holographic film appears random when studied under a microscope. Only when you shine a lazer beam with the same wavelength at the holographic film does the object's form project. Weirdly, if you cut the film in half,  shine the lazer into one half, the projected form has the same image.

The information encoding the image appears to be stored non-locally upon the holographic film, thus accounting for why the patternon the holographic film appears to be random. Interference caused that randomness some how.
Both these systems---holographic projection and quantum fields---involve waves, and interference patterns. Change due to wave propagation seems to be a factor.
