Here's my best analogy: If you take one mole of "random" Rubik's Cubes (the number of molecules in a glass of water), then search for 'Solved' ones you'll find 512 are magically solved, merely by the power of large numbers. The ways molecules can 'snap together' are similarly 'finite', just like a puzzle.
Now think how many glasses of water are on earth. A replicating molecule would have existed probably on the FIRST DAY earth had cooled enough to form water.
If we knew what the shortest sequence if DNA is that can replicate, we'd be able to mathematically state the probability of it existing in any 'random' length of DNA of that length as 1/(4^N) where N is DNA length (number of AT/GC pairs).