In The Existence of God, 2nd ed. (hereafter: EOG) Richard Swinburne defends the following inductive version of the Cosmological argument (hereafter: TCA):

e: There is a complex physical universe.
Therefore,
g: God exists.

A key premise in Swinburne’s (deductive) argument in defense of TCA goes like this:

(TCA9) The probability that there will be a complex physical universe given that God does not exist is low. (EOG, p.151)

I asked Mr. Swinburne to clarify his reasoning on this point, and he kindly provided a detailed explanation (email dated 10/24/11. e means same as above, h means ‘God exists’, and k refers to tautological background knowledge):

Let c be ‘there is a personal creator other than God’. Then (given the sentence on p.149, ‘e could not, as we have seen..’), with k as a mere tautology, P(e &~h&~c&k;) will be the probability that a complex physical universe exists without an explanation. By the calculus this equals P(eI~h&~c&k;) P(~h&~c&k;). I have argued that c is much less probable than h. If (see p.109) P(hIk) is very low, P(cIk) will be even lower. In that case
P(~h&~c&k;) will be close to 1, and so P(e&~h&~c&k;) will equal approx P(eI~h&~c&k;) which – given the very high probability of ~c approx equals P(eI~h&k;) ,which is the probability that there will be a physical universe given that God does not exist. If however P(hIk) were not very low, then the equation would not hold, but I was trying to give the atheist as much as I could, and so assuming that it is very unlikely a priori that there is a God. But if the prior probability of God is higher, then the arguments from e or anything else won’t have so much force, but then they wouldn’t need to in order to reach the same posterior probability. I am sorry that my presentation on p.149 seems to have been a bit sloppy, but – I hope that you will agree – it doesn’t make much difference to the result.
Thanks for your interest – Richard

The overall logic is clear, although a number of details still need to be worked out:

1. P(e&~h&~c&k;) is approximately equal to P(eI~h&~c&k;)
2. P(eI~h&~c&k;) is approximately equal to P(eI~h&k;)
Therefore,
3. P(e&~h&~c&k;) is approximately equal to P(eI~h&k;)

One of the key points here, which clarifies the reasoning for me, is that the probability of there being a complex physical universe that has no explanation is expressed as: P(e&~h&~c&k;).

The idea is that Swinburne has already concluded that there can be no scientific explanation for the existence of a complex physical universe, so assuming that the only other kind of explanation that can be given is a personal explanation (in terms of a creator, or group of creators, who has some purpose or purposes for making a complex physical universe) the denial of the existence of God (~h) combined with the denial of any other personal creator (~c) eliminates the possibility of a personal explanation, and thus there would be no explanation for the existence of the complex physical universe.

In English:

1a. The probability that there is a complex physical universe but there is no God and no personal creator other than God is approximately equal to the probability of there being a complex physical universe given that there is no God and no personal creator other than God.
2a. The probability that there is a complex physical universe given that there is no God and no personal creator other than God is approximately equal to the probability that there is a complex physical universe given that there is no God.
Therefore:
3a. The probability that there is a complex physical universe but there is no God and no personal creator other than God is approximately equal to the probability that there is a complex physical universe given that there is no God.

This would be a deductively valid argument if the relationship was ‘X is equal to Y’, but since the relationship is ‘X is approximately equal to Y’, the inference is not deductively valid, as it stands. I can probably tweak the argument a bit to get it into the form of a deductively valid inference.

There is also another inference needed to get to the intermediate conclusion (TCA9):

3. P(e&~h&~c&k;) is approximately equal to P(eI~h&k;)
4. P(e&~h&~c&k;) is a very low probability.
Therefore,
5. P(eI~h&k;) is a low probability (at most).

Note that (5) is another way of stating (TCA9).