Exactly. So why aren't there more animals alive today with asymmetrical body shapes. Surely it isn't impossible for an animal with an asymmetrical body to survive.
Indeed it's not. However, bilateral symmetry was a trait evolved by an early ancestor of ours, and we've simply inherited it all the way down. Since it evolved so very early on in the history of
Animalia, it forms one of the biggest subdivisions of animals, along with
Cnedaria (jellyfish and the like) and
Ctenophora (tiny animals a few cells thick).
So, why are so many animals bilaterally symmetric? Because it was a trait evolved by one of the first sub-divisions of animals, and has been inherited by its descendants. These descendants include everything from dragonflies to komodo dragons.
Non-bilaterally symmetric animals
do exist (starfish, jellyfish, sponges, etc), but bilateral symmetry is something that evolved so early on in our history as animals, that the majority of animals have it.
I still also say it's impossible for something based on random mutations to be able to produce anything even close to symmetry. To believe that is equivalent to believing my coin flip analogy is possible. 10 may be possible after a thousand tries, but it would be more accurate to use a minimum 100 coin flips for the example. As in, it could never happen.
Your analogy leaves out two crucial components of evolution: reproduction, and natural selection. Evolution can do what it does because a) species reproduce, b) reproduction causes variation, and c) natural selection favours some variations over others. Your analogy only factors in randomness, and falsely assumes that genes are composed wholesale in one grand swoop by a random coming-together of chemicals. They're not.
If you were able to flip 100 coins, then flip 100 sets of 100 coins, and pick which of those 100 sets most closely matches the original 100, hold their similarities constant and flip whatever coins are left (in other words, there's a selection pressure that preferentially selects some coins over others to reproduce into the next generation), then that'd be more accurate.
And, of course, the left and right hand side of the body don't match up by sheer chance, there's essentially a genetic mechanism which says "and now do exactly the same on the right".
My snowflake analogy holds here, too: the water molecules on one arm of a snowflake have no idea what the molecules on the other arm are doing, yet they form beautiful hexagonal symmetry, all by the sheer random motion of energetic atoms. How? Because they all work in the same conditions under the same rules.