This is one of the few topics where I feel everybody's wrong. The fine-tuning argument is wrong because it implicitly makes unreasonable assumptions, but the counters to the fine-tuning argument are completely missing the point and are attacking perfectly valid arguments in the process

(just to clear things up, I'm not talking about arguments that dispute the 10^-10^23 figure - that is not my field of expertise)
I'm going to have to show some probability basics:
P(X) = a priori probability of X: if you close your eyes and ignore all evidence, how probable is X?
P(X|Y) = conditional probability of X, given Y: if you look at evidence Y, how probable is X?
~X = not X
Bayes' law:
P(Y|X) = P(X|Y) P(Y) / P(X)
So here are the premises of the fine tuning argument:
a. P(Life|God) = 1: if God exists, then life is the result.
b. P(Life|~God) = 10^-10^23: if God does not exist, then life occurs with probability 10^-10^23.
Here is the conclusion the argument aims to prove:
c. P(God|Life) >> P(~God|Life): given the evidence of life, it is much more probable that God exists that he does not.
The problem: the data we are explicitly given is insufficient to calculate P(God|Life). Using Bayes' law we have that:
P(God|Life) = P(Life|God) * P(God) / P(Life) = 1 * P(God) / P(Life)
P(~God|Life) = P(Life|~God) * P(~God) / P(Life) = 10^-10^23 * (1 - P(God)) / P(Life)
So we want to show that:
P(God) >> 10^-10^23 - 10^-10^23*P(God)
P(God) >> 10^-10^23 / (1 - 10^-10^23) = (approx.) 10^-10^23
Therefore, in order for the fine tuning argument to be valid, we have to show something awfully specific: namely, that the
a priori probability God exists (and remember,
a priori means we are not given any evidence at all - it is a blindfold probability!) is much greater than 10^-10^23.
This is important, so I have to repeat it: the
necessary and sufficient condition for the fine tuning argument to be valid is for P(God) to be greater than 10^-10^23. That is cold, hard, definitive fact. It comes directly from theorems from probability theory. Period. And if you think P(God) doesn't make sense, because as I said, it's an evidenceless probability, then tough luck - the argument cannot be valid without it and the debate is over.
Now how do you evaluate P(God)? What's its value? Obviously, you must think it's a pretty good probability - I suppose that religious people in general think the same. I even suspect that most atheists have a similar intuition (and that is why they elude the question). Eh, I'm afraid you're wrong. The usual way we evaluate a priori probabilities of this kind is that simpler solutions have greater probability. God is complex. Awfully complex. A good measure of complexity is information content. If we suppose that, for example, God uses a yottabyte of information, its a priori would already be as low as 10^-10^24 (if my calculations are right). Now, if you take a simple program that tries an universe, looks if it could lead to life, then tries again until it succeeds, it would easily fit in a megabyte, and its a priori would be 10^-10^6, which is significantly more likely than both dumb luck AND God.
You may disagree with how I get my numbers (I assure you that I'm not pulling them out of a hat - there are good reasons behind them), but now, at least, you know exactly what there is to argue. The intuition behind the reasoning has merit, but the argument is based on a fishy premise, the implicit assumption that God's absolute probability is greater than 10^-10^23 and furthermore that there are no simpler processes that have higher probability.
PS: you said: "1/1x10^10^23. That’s 1 times 10, to the tenth, to the 23rd", but the former is 10^10^23 whereas what you said in words is (10^10)^23 = 10^230, which is much, much, much, much smaller (thus bigger in probability) - so much in fact that there is no way you will ever convince me that God is more probable than that. So which one is it?