Okay, once again, biochemistry is decidedly NOT chance. Assigning odds is meaningless. But let's play your game anyways.
I don't know why it is assumed that you need 500 amino acid chain. Modern abiogenetic theory supposes that the first proteins were something like 30-40 units long. These would then begin working together, and eventually form into simple organisms.
It can be supposed that the first living things would be similar to the Ghadiri group of self-replicating peptides, or a self replicating hexanucleotide.
Another view is the first self-replicators were groups of catalysts, either protein enzymes or RNA ribozymes, that regenerated themselves as a catalytic cycle. An example is the SunY three subunit self-replicator. These catalytic cycles could be limited in a small pond or lagoon, or be a catalytic complex adsorbed to either clay or lipid material on clay. Given that there are many catalytic sequences in a group of random peptides or polynucleotides it's not unlikely that a small catalytic complex could be formed.
These are not mutually exclusive. Both could have happened.
Lets' assume the Ghadiri group, these self-replicating proteins are 32 amino-acid long chains. Nothing like the 500 units long. This would, by the maths your source has proposed, Starman, have a 1 in 4.29 x 10^40 chance of forming randomly. This is hugely lower than the 1 in 10^600 that your source would have us believe.
Still, that's a HUGE number, and probably quite unlikely. Well let's look at the Ocean size of the Early Earth. This is commonly estimated to be about 10^24 liters. Let's say the concentration of amino acids in this ocean is 10^-6M (moderately dilute on the spectrum of values that it is expected to be). This means we have 10^50 starting chains. This means that we would have 10^31 proteins in under a year. Now, given a million years, there is a good chance that one of these proteins (that are still being made through more time) will be the self replicating one that we are looking for.
But of course, there is more. There are a huge number of other simple proteins that are self replicating that could have started life. We have to take these into account. These increase the odds even more.
As you can see, life is certainly feasible given the size and properties of the early Earth, even using the pessimistic numbers that I have used here.
I can add some scientifically peer reviewed sources for numbers and things, if you would like.