The statement on Lebesgue sets and zero probability is indeed correct, and I have to admit that I got a little sloppy in my explanation of real numbers and zero probability. But then again I finished Real Analysis class a long time ago (I still get nightmares sometimes!) and didn’t quite expect it to creep up on me here. Nevertheless I should have explained the zero-probability paradox straight up and not dumbed it down. Good catch there!
I'm glad to see that my earlier statement that your application of Bayes' theorem cannot be used to give evidence for or against RTB's conclusions is well received. It's clear to me from your explanations that there is a legitimate attempt to show that the RTB use of probability is in need of correction, and that's fair enough; good scientists try to rigorously test theories to see if they withstand cold, hard facts. I will try and show the same spirit of scientific pursuit in my discussion.
However, the reason that I posted the comment was to give a hint as to why you are using Dr. Ross’ applications of probability incorrectly. I thought it would be enough for you to catch the error, but I guess not.
So here I will show my take on your conditional probability argument. And not a moment too soon! I see that the use of Bayes’ theorem is the meat of your argument, and you are right to say that nobody really tried taking it on. But before I do that, I need to point out one critical thing:Dr. Ross has never in his writings said that the probability of our existence is low. In fact, he argues that because the universe was made for humanity, the probability of our being here is actually quite high. He does argue, however, that if there were no God, then the probability of our being would be very, very tiny. Case in point: see the statement you quoted: Dr. Ross said that WITHOUT DIVINE MIRACLES, the probability is 10^-282, making this figure CONDITIONAL ON THERE BEING ONLY UNINTELLIGENT AND UNGUIDED PROCESSES AT WORK.
So with that in mind, I will create the setup of the problem along with some notation:
The setup is pretty much as you had it before.
-X is the event that humans are here (or would have arrived, it really doesn’t matter what time frame is used) -H is the event that God exists.
-Assume that if God exists, he would have rigged the universe to create humans (so P(X|H)=1, as before).
-Notation is same as before, except I will note the “ compliment of H ” to be H’.
-This setup will assume that the universe follows the standard laws of probability so as to avoid any discussion on their validity.
Before I start, I would like to iterate in probabilistic notation that Dr. Ross’ tiny figure of 10^-282 is not P(X), but P(X|H’) by the above argument.
Now, step by step, my result goes as follows:
Bayes’ theorem states:
P(X|H)*P(H) = P(H|X)*P(X) [equation 1]
An elementary result of conditional probability states that:
P(X) = P(X|H)*P(H) + P(X|H’)*P(H’)
But it is already agreed that P(X|H)=1, so
P(X) = P(H) + P(X|H’)*P(H’) [equation 2]
Plugging [equation 2] into [equation 1] and applying P(X|H)=1 to LHS gives:
P(H) = P(H|X)*[P(H) + P(X|H’)*P(H’)]
Using the result that P(H’)=1-P(H) gives:
P(H) = P(H|X)*[P(H) + P(X|H’)*(1-P(H))]
So now we have that:
P(H|X) = P(H)/ [P(H) + P(X|H’)*(1-P(H))]
As far as I’m concerned, this completes my result, since decreasing P(X|H’) must necessarily increase P(H|X). Note that every step here is done according to well known results of probability theory.
Intuitively speaking, this means that providing additional evidence that it is increasingly unlikely that we came about by random chance always increases the probability that God created us given that we are here.
So indeed, Dr. Ross and the scientists at RTB are perfectly justified in their use of probability to argue the case for a creator.
I hope this is the answer you were looking for.