Richard, this has nothing do with fallacies. But I will entertain this question because this is a good lesson in fallacious thinking for all those reading along.
Modern mathematicians in general agree any number larger than 10 to the power of 70 represents zero possibility.
You must provide a source for this. I can't imagine anyone with a modicum of mathematical intelligence making such a claim. An unimaginably small probability is still a possibility. A non-mathematical term, "statistical impossibility," is used with everyday experiences to mean "so unlikely that is not even worth considering." When dealing with billions of years at a universal scale, this casual usage does not apply.
According to British mathematician Roger Penrose, the odds of the universe existing by non-designed random processes is 10 to the power of 10,123 against.
And what are the odds of the universe existing by a magic being that "spoke" it into existence? The common fallacy is the argument from incredulity ... you can't grasp the improbability of an event so you discard it as a possibility. In addition, this is special pleading , because you demand mathematical odds of a universe, but don't demand the mathematical odds of a magic being who allegedly created the universe.
To illustrate how irrationally impossible this is,...
Once again you demonstrate mathematical ignorance here. You are illustrating how improbable this is.
But the number calculated by Penrose...
I am not sure if you realize that Roger Penrose is an atheist he said so himself. This should just give you pause when the guy who's data you are using, who knows more about his data and calculations than any other person in the world, concludes from this that there probably is no God.
What evidence do Neil DeGrasse Tyson and other scientists have, who are betting against the unimaginably overwhelming mathematical odds against the universe being a result of "random", "unguided", "natural" "processes"?
Once again, you are ignoring the calculation needed to make a probabilistic judgement: that the universe is NOT a result of "random", "unguided", "natural" "processes." Until you have this number, you have only one side of the equation which is meaningless. Provide a source where Neil DeGrasse Tyson has said that the universe is a result of "random", "unguided", "natural" "processes".
Isn't science supposed to be based on verifiable evidence? What evidence do they have and, what testable method are they using to verify it?
I know of no scientific claims that there is no God. I know of many hypotheses and theories that rely on natural causes to explain effects, and each of these theories have their own scientific justifications (hypothesis don't require the same level of justification).
I wrote an article on probability and God, explaining and demonstrating in great detail that these calculations use Bayes' Theorem, which rely on estimates of prior probabilities. Depending on one's theological assumptions, the resulting calculation favor a God or a universe without a God. The bottom line is, math is useless for actually calculating the probability of a god, but it is a useful process in that it helps one critically think about all ones assumptions.