[AndrewC]

But if a self-replicating cell of a simpler type can exist (perhaps a lipid membrane enclosing a few protein or RNA fragments, 1/1000 the complexity of the simplest currently feasible cell, and the “laboratory,” instead of ~1 cubic meter is instead the world’s oceans (1,370,000,000 cubic kilometers, or 1,370,000,000,000,000,000 cubic meters), and the “experiment,” instead of taking, say, 10 years, takes 4,000,000,000 years, then the process becomes 5.48E+29 (548,000,000,000,000,000,000,000,000,000) times more likely to produce life.

Since my calculator cannot display all of the numbers involved I must use logs. First 4²⁵⁰⁰⁰⁰ is not a small number, in terms of 10 it is 10^150514.9978 . Consequently even if the odds were increased 5.48E29, ----no let us say 5.48E29 times a billion, billion, billion, billion, billion, times more (or 5.48E74) the odds would be ~1 in 10¹⁵⁰⁴⁴⁰

Since this is beyond comprehension if we use instead 4^250 as the beginning odds, the final number becomes a more reasonable looking 1 in 5.97334E+75, that is including all of the adjustments increasing your odds. Now how does this relate to anything we can conceive? Well given a 20 billion year old universe(fudging in your favor) with 366 days per year(again fudging in your favor), 24 hours per day (no fudging), 60 minutes per hour(again no fudging) and 60 seconds per minute(straight as an arrow) we can calculate that there have been 6.32448E+17 seconds since the beginning of the universe. Hmmm. If we allowed your total scenario to occur every second since the beginning of the universe we cannot cover all of the possibilities for even a chain of 250. Well how about doing the process every picosecond? That would make the iterations 6.32448E+29, still far short of the 5.97334E+75 combinations remaining.

[AntiGuv]

The mathematical odds are of no great consequence unless you have some genuine standard of measure. If the odds are even so monumental that the event would happen only once in the entire history of the universe - and even then, perhaps, by happenstance - then whatever sentient beings arose would still have arisen due to those slimmest of conceivable odds. Until you can assert that it has happened dozens or hundreds or millions of times on other planets, then all you have is the fact that it happened once. However slim the odds of life arising, from whatever source, they became a reality at least in this one instance here on Earth. It does not follow that the evolutionary model is suspect or that natural selection does not occur.

[DWPittelli]

Your math may be correct, but it is not relevant unless we assume that only a single one of the possible chains of 250 amino acids would work. Perhaps many would work. Perhaps all would work. And perhaps the simplest self-repicating structure had no DNA at all.