What is the Probability of God's Existence?

There’s been a few posts recently about using probability theory to illuminate questions of biblical historicity. It reminded me of a review I wanted to post on last year, but didn’t get time for.

In this review (via clayboy) of the new “Agnosticism: A Very Short Introduction”, Vernon paraphrases one (among many) of the book’s arguments on the probability of the existence of God:

Atheism is not the default position when it comes to God – as if the burden of proof was on the theist – for a number of reasons, but most interestingly because atheism is not the most likely as a prior probability (that is, the probability before you consider any evidence.) Anthony Kenny has offered one argument as to why: atheists have to deny every definition of God; theists have only to affirm one. So the atheist has to prove more than the theist to ensure their position stands, which suggests atheism is a lower prior probability.

The use of probability to talk informally about the existence of God has a long pedigree, but a rather ignominious one. And indeed folks like Swinbourne, and Tony Kenny have dabbled in making it a little more explicit. But this specific argument, whilst also being methodologically confused (it doesn’t understand what ‘prior probability’ means, technically) is also naive in a very freshman-probability-course way.

What is the probability of you not rolling a 6, on three fair dice? It is 5/6 on the first die, 5/6 on the second, and 5/6 on the third, giving 125/216 of not rolling a 6. So even though the chance of rolling a six is small, with three shots at doing it, the odds of avoiding a 6 come out at not far from 50:50 (if we had a fourth roll, the chance would drop below 50:50).

This is Kenny’s hypothesis – what is the chance of no God existing, of atheists being right. We calculate it in the same way: What is the change of Yahweh not existing? What about the Christian Trinity? Vinshnu? Zeus? Allah? Isis? Each probability might be close to one: each God may be unlikely. But multiply them together and the chance of none of them existing drops way down. For example, if we had one thousand potential Gods, each with only a one-in-a-thousand chance of existing, the likelihood of the atheist being right is only 35%.

Unfortunately this line of argument is crock. It assumes independence.

When you first learn statistics and probability, chances are your instructor would have spent a lot of time telling you that things are independent. Flipping heads 10 times on a coin does not mean the next flip is more likely to be tails*. This is counter-intuitive, so it gets a lot of emphasis.

But unfortunately, things in the real world are very rarely independent. The independence isn’t normally the kind that we intuit (our intuitions really are most often wrong in probability**). But we cannot assume independence in general. And in my experience even graduate students of science find it hard to shake the idea, planted in high school, that every probability is independent.

The non-existence of Gods, for example, is highly dependent. And without independence, the Kenny system is laughably naive (and a good way to flunk undergraduate statistics). Yahweh isn’t a completely new option to Allah, or even Isis. The mythologies are connected, deeply, historically and psychologically.

Taking the probability of each of a thousand broadly similar Gods as being one-in-a-thousand, the final probability of there being any God, would end up being pretty much still one-in-a-thousand. And that is definitely not a good argument for agnosticism.

There are better ways to argue for agnosticism, I think, and in fact in Vernon’s article he goes onto a much better one, that the existence of God reduces the number of explanations, which in general tends to be associated with better explanations. I disagree with that, too, but that’s a topic for another day.

* I’ve said before on this blog, that the correct answer is that, a coin that flips heads ten times in a row is more likely to flip heads on the eleventh. When I said that before, I was disagreed with. But, nevertheless, I was right 😉

**Here’s a famous example: My sister has two children, one of whom is a boy – what is the probability of the other child also being a boy? (assume that there is a roughly 50:50 male/female split among children as a whole). For the answer, drag your mouse over this invisible text to select it and read it: There is roughly a 1/3 chance the other child is a boy – if that matches your intuition (and you haven’t heard the puzzle before), then you’re very, very unusual.



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19 responses to “What is the Probability of God's Existence?

  1. Hi Ian,
    Yep -I agree entirely. Very good points; I think many atheists would allow a little agnostic wiggle room because it’s not possible to *prove* the non-existence of every class of god. Although Al Plantinga comes close (paradoxically) in his famously inane balls-up of the Ontological Fallacy to disproving the god of classical theism.
    The “other child” puzzle is beautifully counterintuitive. It’s up there with the Monty Hall problem.

  2. Ian

    Yes, the “other child” puzzle is a beautiful case study. As a grad student (in the category I mentioned: conditioned into believing everything is independent), I spent a whole day really understanding it, in slightly broader terms. Specifically what is says about information and knowledge generally:

    As I’ve presented it the answer is 1/3. If I identify the children “my sister’s eldest is a boy, what’s the probability the younger is too” or “my sister’s tallest is a boy…”, then the probability ups to 1/2 immediately. And if you provide additional, but still indeterminate data about the children, it is some way in between: “my sister has a boy born on a Tuesday, what’s the probability her other is a boy?”, for example (the famous Tuesday’s Child problem) gives you 13/27, just under 1/2.

    You know those moments when you just ‘get’ something, and it seems beautiful? It took me some wrangling (and lots of probability tables) to realise you have a certain amount of information about a system – if you can determine a subsystem that you have more knowledge about, then your knowledge about the remaining subsystems drops. If you can show that your knowledge is *only* about one subsystem, then your knowledge of the rest is just the ground probability distribution. Obvious when I write it like that, but hard won, to me.

  3. interesting stuff! i’m bad at math, so this helps as a lesson both in math as well as elementary probability.

    i find “But unfortunately, things in the real world are very rarely independent. The independence isn’t normally the kind that we intuit (our intuitions really are most often wrong in probability**).” rather ironic. this is an essential doctrine of many religions, esp. Christianity. the desert fathers emphasized humility and the Franciscans came along and stated “don’t even be humble, you’re not even THAT important.” trying to show that we are dependent on the wider world, on each other, and on God.

    i wrote in a sermon once about how the Gospel often overturns our natural intuitions and ways of being in the world. i stated “When Jesus told his parables to the people, his disciples asked, why do you talk to them in riddles? And his answer was: “So they won’t catch on. Because anything they could catch on to would be the wrong thing. As Isaiah said, seeing they don’t see and hearing they don’t hear, neither do they understand [Matthew 13:10-17]. That’s why I talk to them like this: because I don’t want them to have little lights go on in their heads. I want to put out all the lights they’ve got, so that in the darkness they can listen to me.”

    that seems to be what you’re saying with probability… but you’re using the analytic method and i’m using the literary method to say something similar. what i’m interested in is what is the probability of a God when the basis is infinite? i guess a similar question would be “what is the probability of life in an infinite universe?” apparently pretty good since we’re here asking the question but this assumes an infinite universe and stems from the bias that i’m on a planet with life on it. my intuition is often wrong which is what my faith tells me and you’ve just confirmed; so what is the probability of both life and God in an infinite universe? do you know or can calculate?

  4. Ian

    That’s a very interesting interpretation of Matthew 13! I might do a post on that – it is one of those problematic verses that it is good to get us all stewing over! I haven’t come across your version before.

    The last paragraph: the probability of life, regardless of the size of the universe is 100%: because it forms part of our ‘ground’ knowledge. It is only possible to form a prior probability if we can estimate the total number of possible universes in which life might exist, divided by the total number of possible universes. Given we have experience of one in both cases, we just simply can’t give those numbers. We can’t even begin to imagine what the parameters of something being ‘alive’ might mean outside the history and physics of our universe. Most people who do this, assume life is something like life on earth, and they get small probabilities. Which is like being asked ‘what is the probability someone will win the lottery this year’, and doing the calculation ‘what is the probability that I will win the lottery this year’.

    So the probability of God existing in a universe: the same math holds, but again I just can’t see there can be any confidence at all in enumerating the number of possible universes containing a God.

    One of the problems with probability is that it appears to be a really flexible way of thinking about the world. And a really simple way of quantifying those thoughts. But unfortunately it is *really* finicky. You get far more garbage out of a calculation than you put in, typically. Small inaccuracies in your assumptions can lead to big changes in outcome. So ‘estimated’ probabilities are just about useless. The idea, as some have had, of using Bayes theorem to estimate the probability of the resurrection, is an example. All you get out in those cases is a numerical estimate of your prejudices, not any information about reality.

  5. wow! that’s a lot to chew on. what i get out of all that is “the question and they way it is ask is just as if not more important than the answer.”

    Richard Rohr states throughout his book “Things Hidden: Scripture as Spirituality” that “how you get there determines where you will finally arrive. the process itself is important and gives authority to the outcome.” is assumption about the bible and religion is also mine in that the text mirrors human consciousness itself as it includes both the development of prime ideas (universality, servant-hood, radical self-sacrificial love, care of the oppressed/widow/orphan/etc) as well as the fight and resistance against those ideas. it is faith and unfaith locked within the text. it is a three-forward-two-backward step text just as humanity is, IMHO.

  6. Ian

    Yes, that’s a fair take away.

    In the case of asking questions about probability, I think we have to understand that there really is reality. And getting out what you put in, does not mean that what we all get out is equally valid! Having your prejudices reflected back to you is not the same thing as learning something about reality.

    I like the precis of the book though. I think the balancing act between fighting the known biases and frailties of human mind, and enjoying the experience of having a mind, is a core part of the human condition.

  7. Boz

    I responded to this previously on a different blog, but didn’t get a coherent answer.

    Using your 1000 independent gods example, atheism is 35% likely. However, monotheism (e.g. believing that only Baal exists) is 0.0368% likely.

    This is because the monotheist believes that 999 gods do not exist and 1 god(Baal) exists. While the atheist also believes that those 999 gods do not exist, and also that Baal does not exist. So, under this example, atheism is roughly 1000 times more likely than any monotheism.

    And also with the original quote, the author is ignoring the 999 gods that the monotheist must deny.

  8. @ Ian:
    Great math reminder to help us from being deceived. The great problem with reason, is that it is often counter intuitive. 🙂

    @ Ghost:
    I would suggest not throwing around the phrase “The Gospel” because its meaning changes from Christian to Christian and thus on a non-confessional site [that is, one of your own personal confession], it is best to spell out what you mean. For a fundamentalist Christian, “The Gospel” could mean [that believing Jesus physically raised from the dead by his Dad means you can live forever in heaven unlike non-believers]. “The Gospel” is probably not a clear way to dialogue on an Atheist site unless as an outreach method, perhaps.

    Secondly, when you say “the Bible mirrors human consciousness”, I must object that by your reasoning, any book of literature or even mathematics could do the same. And since all do this, then your statement tells us nothing about the Bible. Unless, of course, you tell us how the Bible is unique in “mirroring human consciousness”. Otherwise, it is can be an empty phrase just like “the Gospel”.

  9. Ian

    Boz, excellent point – if the probability of Gods is independent it is exactly as you suggest: the likelihood of monotheism being right is just about as low as atheism (slightly higher, but ballpark), and the odds are very good of polytheism being correct. If the probabilities aren’t independent (as the vast majority of theists would contend: the fact that Yahweh exists means that Thor doesn’t) then the probabilities are dependent and the probability of any of them drops to almost zero. Hoisted, as they say, by their own petard.

  10. John Clavin

    Maybe off topic, but on the subject of probability and god, I would say that “god” is probability. And the lack in understanding of probability is the basis and reinforcement of the religious/god model.
    A person that wins an impossible to win lottery, but doesn’t understand the probability of large number sets, will conclude that god answered his prayers, or some people think that life on earth is impossible without divine intervention because they don’t understand the math involved. My neighbor got a residual check in the mail right when she really needed the money, so “there has to be a god.”
    I would say that the religious concept of “faith” is all about people trying to understand the probability of events that shape our universe.

  11. @ Sabio: “I would suggest not throwing around the phrase “The Gospel” because its meaning changes from Christian to Christian and thus on a non-confessional site [that is, one of your own personal confession], it is best to spell out what you mean.”

    Gospel = Four books found towards the end of a bigger book called the bible. also means ‘good news.’

    “Secondly, when you say “the Bible mirrors human consciousness”, I must object that by your reasoning, any book of literature or even mathematics could do the same. ”

    yup, everything mirrors human consciousness mostly because we’re the one’s finding meanings and patterns in things. i think the bible mirrors it as it’s a ‘text in travail’ where we see a group of writers from a certain cultural and religious perspective opening up and getting past their tribal lines and then withdrawing and damning everyone outside their tribe. kinda like what you just did here… “listen here Christian, know your place!” and then proceeded to knit-pick my comment apart and declare it meaningless. to me, that mirrors our consciousness which mirrors what i see in the biblical meta-narrative. hope that clarifies.

  12. Quantum137

    The givens are that consciousness exists, and that we do not have a precise or operable definition of consciousness. We assume that consciousness arises from or is an emergent property of matter. At the same time, certainly from a quantum mechanical perspective, we cannot make a clear delineation between consciousness and matter as a result of the behavior of matter at the quantum level or on the basis of semantics. If we assume that consciousness is separable from matter the task is easier and overall likelihood increases, but our calculations do not depend on this separability.

    Given those assumptions we can proceed with possibilities, for which we need not assign probabilities as these are assumed to be very, very small. The law of large numbers will take care of this as will be explained.

    As it is possible that every particle of matter exhibits consciousness, then more complex arrangements of matter exhibit greater degrees of consciousness. From this we can reason by analogy that if our brain is a mechanistic computer, and possesses consciousness as an emergent property, then the universe as a whole could possess a consciousness. Some have hypothesized (Seth Lloyd) that the universe itself is a giant computer. Our ability to comprehend that consciousness would be an equivalent feat to a single neuron being able to comprehend a human brain.

    If some theory of multiverses (e.g. many worlds) is correct then with each choice that occurs a new universe is created, an uncountable number from moment to moment. These choices can be the result of our own conscious choices, on the macro level, or the result of a photon passing through one slit instead of the other.

    If true, then one can suppose that countless universes have been created “in the past.” Universes may have been created with vastly different physical laws than our own. Theorists suggest there could be parallel universes co-existing with our own, yet we are totally unaware of them.

    Over countless path divergences one can calculate that, although a vanishingly small probability for a single universe, eventually a universe would evolve that only or primarily consisted of a consciousness, called “C.” Through a process roughly analogous to biological evolution and life arising through abiogenesis, some given C universe could itself be capable of reproduction in the sense of being able to create a new universe consciously (via the multiverse principle or a variation thereof) which we will call “U.” Perhaps there is some underlying property of entanglement or D-brane like interaction that is totally novel that would allow C to co-exist with U with an “awareness” of U and some unspecified ability to interact.

    C could legitimately claim to be the creator of U, theoretically with powers of omniscience, omnipotence, and omnipresence with respect to U. Conscious residents of U could legitimately claim that they live, move, and have being with respect to C, without understanding how C interacts with U.

    The overall probability of C critically depends on the number of universes, N. From the foregoing we can make a rough estimate of the probability of C as proportional to 1-(1/N).

    This, then, is the theotropic principle.

  13. Ian

    At the same time, certainly from a quantum mechanical perspective, we cannot make a clear delineation between consciousness and matter as a result of the behavior of matter at the quantum level or on the basis of semantics.

    I didn’t understand that at all. ‘Quantum mechanical’ does not mean mystical. What specifically about quantum mechanics prevents you from distinguishing between the physical behavior of matter and semantics?

    I should warn you that there are more than one of us here who actually understand real QM. A little-known-fact, my mum’s research was in Quantum physics.

    If we assume that consciousness is separable from matter the task is easier and overall likelihood increases, but our calculations do not depend on this separability.

    Why? In what way and in what possible way would assuming that consciousness is separable from matter make the likelihood of anything increase?

    You seem to be saying

    1. Assume that consciousness is a thing.
    2. Assume that everything in the universe has it.
    3. Assume that consciousness is accumulative over groups of things.
    4. Therefore the whole universe has it maximally.

    You don’t need your multiple universes or the ability of the consciousness to create. That’s just building higher onto dubious foundations. And then you go into the whole ‘theoretically with powers of omiscience, omnipotence and omnipresence’ stuff, which is just hand waving nonsense. And then try to bring it back to an equation of probability!

    Sorry, but there is no theory more abused for doing bad philosophy than QM.

  14. I’m with Ian on this one; I had meant to post a reply, but hadn’t got around to it. No theory in physics has been more roundly abused by crazytalk than quantum mechanics. I would go as far as to say that there is nothing *really* strange going on with QM – it’s just that we don’t understand it. But it certainly isn’t a gap that we can squeeze space pixies or even consciousness into. Roger Penrose got a lot of coverage (and a fair bit of flak) over “The Emperor’s New Mind”, which is a very good book, but his idea of quantum consciousness is built (I suggest) on an incorrect premise and a wrong evaluation of how the human brain actually works.

    I agree with Ian that assuming that consciousness is “a thing” is by no means a given – I can’t even tell whether I was conscious 2 minutes ago. My brain seems to suggest that I was, but I have no way of verifying that, and even if I *wasn’t*, my brain would still say that I was! Oh noes!

    FWIW, the evidence seems to suggest that “consciousness” is indeed an emergent phenomenon, but it’s a phenomenon that is based on many many inter-relationships between “computing” nodes (in the case of the brain, neurons). If (and this seems likely) we were able to fully simulate a neuron, we would be able to fully simulate two interacting neurons. And three, and four and 10 trillion – limited only, really, by our compute power. However, this *necessarily* means that the simulated consciousness is a purely mathematical entity, as mathematical as a cube or the Mandelbrot set, or the Fibonacci sequence. And as such, despite it being hosted on our compute system, it cannot be said to be *dependent* on that system for its existence, any more than the Mandelbrot set needs a “computer” to “exist”.

    The question, then, is whether our entire Universe is not such a mathematical entity. Max Tegmark thinks it is, and I agree. Mathematics *itself* is the fundamental layer of reality.

    The implications for god are pretty obvious – a god *cannot* be a more basal layer of reality than mathematics itself. If it could change the value of Pi, that might be another matter. But it can’t, can it?

  15. Greg

    The probability of God existing is zero. It is impossible for there to be a god who has been around forever and who also created everything in the universe. If there is a god, he must be made of something, otherwise he wouldn’t exist. Therefore, this substance of which he is made, whatever it is, must also have existed forever, which means that God didn’t create everything in the universe because he couldn’t have created the very substance of which he is made. If this substance has existed forever then the rest of the universe could have existed forever. Therefore, there is no need for a god.

  16. Ian

    Thanks Greg, welcome. That’s a interesting logical argument, but not a probabilistic one.

    Like any argument it has its axioms, and I suppose believers would contest those. But I’m not a believer, and I tend to say that I believe there is no God (i.e. a positive statement, not just that I don’t believe there is a God), for similar reasons: I think most concepts of God are self-contradictory. But not all. One thing about humans is they are endlessly creative, particularly when it comes to creative rationalizations for what they want to believe. So I suspect arguments from logic (like yours) or probability (like the one in this post), don’t do anything much other than make people feel that they are even more right than they previously thought.

  17. John Clavin

    I have to agree with Greg on this. The only answer is zero probability. If there was intent to create existence then that intent had to come from somewhere and we would have turtles all the way down.
    There is a great scene at the beginning of “Star Trek Into Darkness” (2013) where the primitive tribesmen find their God.

  18. Bruce

    I wondering pondering this same question…..after speaking to many friends and colleagues, very well educated and logical people. Quite a number of them claimed to have interacted with ‘spirits’ or dead relatives, a friend and his brother and neighbor all claim to have interacted with a ‘spirit’ together at the same time.

    I personally haven’t had this experience yet….but I’m surprised at the number of people in my sphere of activity who have.

    So are there ‘super natural’ beings out there? If so, are they dead humans who have evolved to a higher form of life? Aliens?

    It seems these creatures, spirit or otherwise seem very reluctant to make themselves publicly available on request.

    BTW a mathematician has already worked out this threads topic.


  19. Ian

    Thanks Bruce, and welcome.

    Yes its odd not only that spirits are difficult to pin down, but that they also appear and do things that are largely what you’d expect them to do if they didn’t exist. So if you start from the idea of spirits being imagined, and figure out what kinds of experience you’d expect, it isn’t far from the truth.

    As for the ‘mathematician’, well, I guess I qualify as a mathematician too, and I say that Bayes Theory is renowned for telling you exactly what you want it to. With it people have shown the resurrection is a historical certainty, that Jesus never even existed, that God very likely exists, and certainly doesn’t, and many more. At the point when someone says they used Bayes’s Theory to give a theological, historical, or sociological result, smile, nod and phone the men in white coats.

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