Hmm - I’m not sure that Paul has the weakest grasp on probability theory in this discussion.
You shuffle a card pack, and what is the probability for the sequence dealt? That depends partly on how many cards, which dictates the possible number of combinations and, yes, because that is 52 factorial, it comes out as around 8X10^67, which rivals the number of particles in the universe in some estimates.
But that only translates into a probability as it is used to measure our ignorance of the actual causes involved. There is no absolute probability in existence at all, and no practical probability without knowing some of the specific conditions. In this case we know a lot more than you’d think - that there are 52 cards which, when turned over, will each be different. That gives us a basis to calculate the total of combinations possible, ie that 8X10^67. But we’re not there yet.
Our shuffling involved a set of movements which determined exactly which sequence will appear once we turn the cards over, and given high speed photography and a knowledge of the original order, that sequence would be knowable. So the probability for the machine operator to predict the outcome is 1. But the shuffler, if he played fair, has only an 1:8X10^67 chance of predicting the outcome.
So which probability is “right”? Answer: the question is meaningless, being entirely one of knowledge or ignorance of prior causes - the actual order is fixed from the moment you stop shuffling. If you sneakily saw the first cause, the probability will increase to 1:51 factorial.
And of course, if you were a card sharp and chose the sequence, the probability for you is 1, but for me, even if I know you did it, still 8X10^67.
Conclusion: there are no absolute “probabilities” in the real world, but only events about which, when we know only some of the causes, we can measure our ignorance of the rest by a probability. If we know none of the causes, no probability exists, any more than you can measure the logic of an unspoken sentence. It’s not that we can’t calculate the probability - there just isn’t one there to measure.
So what of an event that is said to be a one-off (I’m British, you see!) - let us say the origin of life. How unlikely is that? Since we cannot replicate the causes of that event (and are actually ignorant of them), we don’t even know the “number of cards”, so are in no position to talk about “probability” at all.
If we imagine life must hinge on some individual chemical reaction, I suppose we could do some kind of estimate of the total possible number of chemical reactions, maybe. But it would be as meaningless as Drake’s equation, because our “chemical theory” is something we made up, not real knowledge of what began life. We don’t even have a secure handle on what constitutes life. Our probability calculation is, essentially, a fairy tale… though if our chemical theory happened to be correct, to an omniscient God, the probability of that combination occurring would be 1, just like the guy with the high-speed camera watching you shuffle.
Presumably a non-omniscient god would at least know the requirements for life, and so have enough information to do a probability calculation. But who wants to believe in a god with a calculator?
If, instead of waiting for chemistry, God deliberately made the first cell on his celestial workbench, to him the probability would be 1, and our probability would be 1 if we believe God did that, or 0 if we don’t. That’s because he stacked the deck which, in this case, only has one card, called “origin of life”.
As far as I can see, there is no science to be done on such a singular contingency, beyond recording that life exists, so it started somehow. And you don’t need a degree to conclude that. But we can be a little sciency if we have concluded it did, indeed, only happen once.
First we can be pretty sure we’ll never find the cause, since if it hasn’t happened again in 4.5 billion years it must involve absolutely unique circumstances we won’t even guess how to reproduce.
We can also say, with some certainty, that there are not a multitude of possible ways for life to form, of which the one that occurred just happened to be the one dealt by nature. If that were so, “life forming events” would be a class containing many examples we would have discovered, and then we could start doing some kind of scientific probability calculation, having more than one point to work on.
(To argue that the existence of life shut the door to all further such events is another mere assertion for which we cannot even attribute a probability).
Lastly, we can say that we have no natural information about the event that would enable us to say whether it belongs to the category of “fortuitous” and “natural”, or “supernatural.” However, calling it “natural” would be meaningless if it is genuinely fortuitous, as far as doing “systematic science” is concerned. One point and the origin is not a good basis for ascertaining knowledge of cause.