Since you both asked the same question, I'll group the response here. The answer to you both is that I cannot prove a non-zero probability. I do not know for certain that life can arise from non-life. However, I think there are some relevant observations that may yield at least some insight.
It's important to keep in mind that a cell didn't just pop into existence by some happenstance collision of organic molecules. The odds of that occurring would be
virtually insurmountable even for a universe orders of magnitude older than our own. Instead, it is much more reasonable to believe that abiogenesis occurred in a step-wise fashion with each step having reasonable (or at least manageable) probability of occurring. The earliest steps towards life might only qualify as life under the most rudimentary of definitions.
Life does arise from non-life, at least in a philosophical sense. The atoms that make up the cells in my body are not fundamentally different than other organic molecules that may be floating around wherever. It is thus reasonable to suggest that life is not a property of matter, rather it is a property that arises from certain organizations of matter. Life is found in the structure of matter. The question then becomes whether or not matter can be organized into what is termed "life" by some natural process. In other words, is there a path by which non-living matter can be gradually (or spontaneously) arranged into a pattern we would call living. This is still the subject of intense research and debate in the scientific community, but there are a number of observations and hypotheses that are showing promise. For example, we know that RNA can spontaneously form and "survive" under certain conditions (
1). We know that from a random pool of RNA polymers, aptomers and ribozymes can spontaneously form (
2). Some of these ribozymes can polymerize and self replicate (
3). Even more, some ribozymes can catalyze reactions between small molecules and overcome small entropy hurdles (
4). Some can even form peptide bonds between amino acids, thus behaving as a primitive
translation system (
5). So we know that ribozymes, which can spontaneously form, can perform a number of different tasks seen in modern cells: replication, elongation, protein synthesis, and enzymatic activity (protein function). Under this model, the earliest of natural selection took place in pools of RNA -- with random chains spontaneously forming and breaking. The chains that most efficiently catalyzed their own reproduction and were able to latch on to free nucleotides the fastest obviously out-reproduced other chains. We also know that phospholipid bilayers, which form the basis of the cell membrane, form spontaneously. It is reasonable to suggest that at some point a replicating ribozyme found the protection of a phospholipid bilayer and thus gained considerable advantage over other replicating ribozymes.
Now, I don't know
for certain that such a scenario happened. My intent was not to say "look this happened". But I think it's a reasonable model and I think that showing that each of these steps is a possibility might give us clues to whether or not abiogenesis has a non-zero probability. In my mind, the defining moment will be when scientists are able to successfully synthesize such a protocell in the lab. Once there is a proof-of-principle, I think the rest will come much more quickly. To be honest, I don't think we're a long way off from this.