How likely is it that DNA was created by nature?

In other words, what is the (mathematical) probability that given the timescale and world conditions at that time (approx 4 billion years ago) that nature could have come up with (or evolved) a structure like DNA.

Given that DNA is a complex molecule that has a very elegant system for holding digital information and a very reliable system to copy itself, with seemingly just the right amount of mutation to help evolution. Can we show that it might have happened within the timescales available or does in look like that it would be extremely unlikely?

Another thing that puzzles me is that DNA is a digital information system. When we look at many naturally occurring systems they tend to work in an analogue fashion. What are the probabilities of other types of systems occurring, such as for example, a potential differences caused by different concentrations of molecules used as a basis for information storage?

LUCA - understood to be all life's universal ancestor - a single cell about 3.5 billion years ago (I think). This does not leave a great deal of time to have evolved DNA from when the earth could have had the conditions to allow this. And, presumably DNA would have needed to have been around for quite some time before it managed to evolve LUCA.

Are there any (computer) models that identify these probability issues, what work has been done on this - anyone know? By the way this is a layman's question - I am not a geneticist!