Q: How do I estimate the probability that God exists?

Mathematician: Before jumping into this question, it is important to realize that probabilities are not objective, observer independent quantities. We can think of the claim that a particular outcome will happen with a probability of 0.30 as meaning (loosely speaking) that given the information available to me right now, if I could replay this scenario many times, then in about 30% of those occurrences I would expect that particular outcome would occur. Notice that this means that my estimated probability may change if the information that I have changes.

To illustrate this concept, consider what happens when different people have different information about the nature of a single coin. For instance, suppose that I flip a coin, and you have to guess whether the coin lands on heads or tails. From your perspective, you estimate that the probability of a head occurring is 50%, based on what you know about coins in general, and the fact that you have no knowledge indicating that a head would be either more or less likely to occur than a tail for this coin. I, on the other hand, am aware that this is a trick coin with a head on both sides. So from my perspective, the coin has a 100% chance of landing on heads. Little do I know, however, that the coin is in fact a magic coin (the warlock who sold it to me at the carnival forgot to mention this fact) — 30% of the time this two-headed coin is flipped it magically changes into a two-tailed coin before landing. Hence, from the warlock’s perspective, the probability that the coin will end up showing heads is 0.70, whereas from my perspective the probability is 1.0, and from your perspective it is 0.50.

So getting back to the God question, we cannot talk about a single, universal probability that God exists. Rather, this probability will necessarily be dependent on the information that you happen to have.

Another important observation is that it is problematic to throw around the word God relying on the assumption that we all know what that means. How are we going to define God? If it turns out that Zeus exists, would we consider that God? What if our universe was created by a pair of powerful, omniscient, omnibenevolent beings? Would we consider them both God? Or how about if our universe was created by an alien scientist, would we consider that scientist God?

We cannot possibly assign a probability to “God” without specifying further what is meant by this term. Hence, rather than a single God probability, it is more reasonable to consider the probability that each possible god G exists (using whatever definitions for G we care to analyze). This information can be encapsulated by a function P. For each event G_exists, representing the existence of god G, P(G_exists) gives the probability of that event (i.e. the probability that G exists). As we have seen, the function P will depend on all the information that you currently have, which we will call your evidence E. Therefore, it is more rigorous to write this function as P(G_exists|E). This is what’s known as a “conditional probability”, and the vertical bar “|” is typically read as “given”. Hence, P(G_exists|E) can be thought of as “the probability that god G exists given our evidence E.”

Estimating the function P(G_exists|E), which assigns probabilities to the existence of possible gods G, is no trivial matter. For one thing, the human brain misjudges probabilities all the time (See this, that, this other thing and also this for some standard examples). Despite these challenges, we can talk about some useful rules that should be taken into account during this process of estimating P(G_exists|E):

1. If the evidence E that we have is really unlikely to have occurred given the existence of a particular god G (but not so improbable otherwise), then that will tend to make the god G less likely. An extreme example of this is that if G is defined to be “an all powerful god that would never allow human beings to live”, then because our evidence E includes living human beings, that god G can’t exist (so we’ve just disproved a god!). Another example is that if your evidence says that there is a lot of evil in the world, then that is going to make any very powerful god that wouldn’t allow evil pretty unlikely.
2. The more conditions we tack onto our definition of a god G, the less likely it will necessarily be that the god G exists. For example, a god that “is omnipotent and omnipresent” is going to be strictly less likely than a god that is just defined to be “omnipotent” or just defined to be “omnipresent”, since the probability that two conditions are met is always less than or equal to the probability that just one of those conditions is met (and neither omnipotent nor omnipresent implies the other, making this inequality strict). Similarly, a god that “helped the Jews escape from Pharaoh” is going to be more probable than a god that “helped the Jews escape from Pharaoh by parting the sea”, since the latter is the same as the former except with extra conditions. And this is not a statement of opinion, but rather, a consequence of the rules of probability.
3. Make sure that the way you use probabilities conforms to Bayes’ Rule. This mathematical rule tells us that something is evidence in favor of a particular hypothesis if that something is more likely if that hypothesis is true than if the hypothesis is false. So, for example, the fact that canaries exist is (a small amount of) evidence in favor of a creator god that loves canary-like-things (compared to the hypothesis of a creator god that doesn’t love canary-like-things) since the probability of canaries existing is greater if a creator god loves canary-like-things than if a creator god does not.

These rules aside, giving advice about how to estimate P(G_exists|E) is quite difficult. But, I can at least warn you away from a few very common but very flawed methods for estimating the probability of various gods.

Bad Method 1:

Set P(G_exists|E) = 1  for whichever god you were taught about as a child, and setP(G_exists|E) = 0 for all other gods. This strategy works well on average only if  (a) there really is a god and (b) that god is by far the most popular god. Given, however, that there is no religion which more than one third of the people in the world believe in (not to mention the huge amount of disagreement within each individual religion about the exact nature of god), this strategy will assign 100% probability to an actually existent god less than one third of the time, regardless of what the truth about god actually is. (At least, this is true as long as we consider the Muslim Allah to be a different god than the Christian one, and both of them to be different than Vishnu, etc.). In fact, the odds could be much worse, if it turned out that, say, only 3,000 people in the world have identified the one true religion (which implies that you almost certainly weren’t raised in that religion as a child, and hence that your function P(G_exists|E) will almost certainly assign 100% probability to the wrong god if you use this bad method).

Bad Method 2:

Use a definition of god that is vague enough that you yourself don’t have much of a clue what “god” really means (e.g. “god is a force” or “god is that which is good”), and then assign P(G_exists|E)=1 for this vague god and P(G_exists|E)=0 to, well, whatever is not covered by this definition. Definitions like these are just too fuzzy to mean very much. If you try to apply reasoning to fuzzy, ill-defined ideas, you’ll often get nonsense as a result. Case in point, lame attempts such as: “God is good” and “Good exists”, therefore “God exists.”

Bad Method 3:

Assign a high probability to things that appeal to you, and a low probability to things that do not appeal to you. Hopefully the problems with this approach are fairly obvious. You may have had a happy day dream about a guardian angel that is looking after you, you may desperately want there to be such an angel, you may spend hours thinking about such angels, but none of that constitutes useful evidence about whether an angel will actually catch you if you trip and fall down the stairs. You cannot (rationally) believe something simply because you want it to be true. You can only (self-delusionally) believe something because you want it to be true.

Bad Method 4:

Only seek out information that supports your pre-existing beliefs, and ignore or avoid information that might disconfirm your beliefs. In practice, this often amounts to starting with a high probability assigned to one particular god J that you happen to have been taught about, and starting with a low value assigned to P(G_exists|E) for other gods G. You then proceed to only ever read supportive literature (and talk to supportive people) arguing in favor of the god that you already think is likely, ignoring literature and people that discuss why god J might not exist. Of course, this approach naturally will cause you to keep increasing your probability P(J_exists|E) and keep decreasing P(G_exists|E) for the other gods, because you keep inundating yourself exclusively with information that supports what you already believed.

Imagine that someone alive in Greece in the 5^th century BC were to follow this method. For instance, suppose that this person started out with only a moderately strong belief that Zeus exists, but then only ever listened to people talking about reasons why he should believe in Zeus. Because of this, his moderate belief in Zeus naturally would have risen over time until it became a strong belief. But this procedure would cause his belief in Zeus to rise whether or not Zeus exists! Hence, it is not a procedure that produces truth, it is merely a procedure that produces belief. This mistake is very common because people tend to surround themselves mostly with people who believe similar things to what they themselves believe, and members of religious communities work to convince each other to believe ever more strongly. For instance, our Greek friend would have been likely to spend most of his time around others that believed in Zeus, rather than those that were marking arguments against Zeus or in favor of a different set of gods.

In summary, to maximize your chance of believing the truth, you must not assume that what you were taught is necessarily true, you must define your terms as precisely as you can, you must surround yourself with the best possible arguments both for and against a particular belief, and you must evaluate these arguments objectively, without regard for what you want to be true.

You may at this stage be wondering: “How does the religious concept of faith factor into our probabilistic argument?” Well, the tricky part about faith is that, while it’s all very well and good to have faith in a benevolent god who does exist, it’s not a wise idea to have faith in a god who doesn’t exist. (I hope that most of us can agree on that point.) Therefore, one must choose carefully, considering the many mutually exclusive gods out there who are allegedly demanding our faith. Faith doesn’t really get us out of the probabilistic quandary of estimated P(G_exists|E). At best it just changes the probabilistic question from “which god should I believe in?” to “which god should I have faith in?” which doesn’t really help.

Probability theory provides a useful framework for thinking about God, not so much because of the specific nature of the God question, but because probability provides a useful framework that can be applied to nearly all questions of (non-tautological) truth.