Calculating the Probability of God’s Existence
Dr. Craig, I am a long timer viewer, my wife and I personally enjoy your On Guard book and my kids enjoy the What is God Like series. My question is one born from frustration dealing with a particular atheist colleague of mine. I definitely affects others around me, including myself, and I can't find anything in your books or website to help address this regarding the probably of God existing equals zero.
He says that When you want to find out the probability of an event happening, you simple divide the event by the total of all events. A simple example of this is the probability of rolling a 1 on a six-sided die, which would be 1 (the event you want) divided by 6 (all possible events). So when you want to know the probability that god exists, you simple divide the one you choose by all possible other events. Since you have no proof that indicate any one god is more likely than any other god, this gives you an infinite number of possible events. So doing the probability you get 1 divided by infinity which is zero." When questioned that infinity is the incorrect constant to use he replied with, Ok, why isn't infinity the right number to use? Do you have some proof for a specific god that nobody seems to know about? If you don't, then how is any other god not just as likely? This isn't rocket science, it's basic logic. Since I don't have to prove they exist, I can make up new gods all day long. When something is unprovable it has a infinite set of like instances by default. Again, that's basic logic.
While your question raises some technical issues in probability theory, to which I’ll return at the end of this answer, in all candor, Jason, given your familiarity with On Guard and other resources, answering this question should have been easy for you! Look at p. 161 of On Guard:
Probabilities are always relative to some background information. . . . Now the atheist says God’s existence is improbable. You should immediately ask, ‘Improbable relative to what?’ What is the background information? . . . The interesting question is whether God’s existence is probable relative to the full scope of the evidence.
Had you asked that question of your friend, it would have been evident that he is considering no background information at all! He seems to be talking about a sort of absolute probability of God’s existence Pr (G) in abstraction from any background information B and specific evidence E. That’s a pointless exercise. He seems to be imagining all the possible deities that could exist and asking, “What are the chances apriori that a certain one of these exists?” How silly! That’s like inquiring about the absolute probability that a certain person, for example, you, exists, given the infinite number of possible persons there could be. Nobody is interested in such absolute probabilities, if there even are such things. What we want to know, rather, is the probability of your existence or God’s existence relative to our background information and specific evidence: Pr (G|E & B).
Now in On Guard I give you four independent arguments for God’s existence that go to show that Pr (G|E & B) >> 0.5. Thus, you should categorically reject your friend’s assertion that “you have no proof that indicate any one god is more likely than any other god.” (How could you have missed this, Jason? That’s what these arguments are for!) When he demands, “Do you have some proof for a specific god. . . ?,” you should reply, “Yes, I have evidence relative to which God’s existence is quite probable.” Then he has to deal with your arguments. He can no longer appeal to his apriori probability, in the same way that someone denying your existence has to deal with the evidence of your existence. That launches a discussion of the theistic arguments, which is exactly where you want to be!
As for the technical issues, when your friend asserts, “If you don't [have evidence of God’s existence], then how is any other god not just as likely? This isn't rocket science, it's basic logic,” he’s presupposing a theory of logical probability which is highly controverted and is rejected by almost all probability theorists today. I ran into this question in preparation for my last debate with Michael Tooley several years ago. I consulted Timothy McGrew, Professor of Philosophy at Western Michigan, who writes on probability theory. Tim explained to me that most theorists would deny that in the absence of evidence the probability of God’s existence is to be computed as your friend suggests.
It is correct that in the complete absence of evidence there is a sort of symmetry of ignorance about competing views. We’d have no idea which is true. But your friend interprets this to mean that the competing options are all equally probable. And that’s false. To see why, consider an illustration provided by the mathematician Peter Walley of a closed bag of colored marbles. If you reach in and draw a marble, what is the probability that the marble will be red? Walley says,
A naïve answer is to say that, because there are two possible outcomes (red or non-red) and no information to favour either, the probability must be 1/2 . . . . But one could apply the same principle to the colors blue and green instead of red . . . and they cannot each have the probability 1/2 . . . . Any precise assessment seems quite arbitrary.1
According to Walley, the correct answer is to say, “I don’t have any information at all about the chance of drawing a red marble, so I do not see why I should bet on, or against, red at any odds.”
Wally then provides a different model of probability which assigns, not precise values to different alternatives, but intervals. For example, in the absence of any information about the color of the marbles in the bag, the model assigns the vacuous probability of 0 to 1 of drawing a red marble, which is just what it should be for a state of complete ignorance.
Applied to the existence of God, what this means is that in the absence of any evidence whatsoever, we should simply have no opinion about whether or not God exists. There is no implication that the probability of God’s existence is 0.
Your friend’s theory resembles Rudolf Carnap’s Logical Foundations of Probability (1951), in which Carnap attempted to formalize prior probabilities in terms of state descriptions and structure descriptions of a system. McGrew comments,
The attempt to nail down prior probabilities in an objective manner using state descriptions and structure descriptions does capture two of our intuitions: it permits learning from experience, and it endorses the commonsense idea that in the utter absence of information, it would be rash to be very confident of a complex contingent claim. But it also has many problems that have been well known since the publication of Carnap’s Logical Foundations of Probability in 1951. In particular, the probabilities are relative to the language used in the description – adding more predicates changes the probabilities, a fact that Carnap himself understood very well. There are other approaches to learning from experience that do not suffer from this defect. To use this sort of artificial system to raise a presumption against the existence of God is really rather comical.
So don’t be fooled by your friend’s confident appeal to “basic logic.” He’s just blowing smoke.
 Peter Walley, “Inferences from Multinomial Data: Learning about a Bag of Marbles,” Journal of the Royal Statistical Society B 58/1 (1996): 3-57, pp. 4-5.