If I experience a coincidence or a coincidence happens in the world that seems to be at extremely low odds, does this imply that God exists? If it doesn’t imply that God exists, can it at least make it more likely that God exists?

Suppose for example that a person is standing on stage and says “God, if you exist, strike me with lightning right now” and a lightning strike occurs that barely misses them, is this evidence of God? Can it make it more likely that God exists?

How should you update your credence in a deity based on an event like this? A skeptic may say “why didn’t God just hit them directly?” A religious person may say “the chances of even that were way too low. This must be divine.” What should the correct reasoning be here?

  • 2
    The correct reasoning is not to waste time on individual events with infinitesimal evidentiary value. One needs a massive number of "coincidences" to get anything tangible, until such dataset is assembled and systematic confounding factors are accounted for fussing over probabilities is pointless.
    – Conifold
    Commented Mar 16, 2023 at 7:22
  • 7
    Two fallacies: a) Non sequitur: a coincidence is as much an evidence for a god as an evidence for unicorns b) Fallacy of Probabilities: low odds does not imply impossibility; there will be always someone who wins the lottery against all odds.
    – RodolfoAP
    Commented Mar 16, 2023 at 8:25
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    @AgentSmith: What you call "deeply-ingrained intuition" might be described as "brainwashing" by others. Commented Mar 17, 2023 at 6:54
  • 3
    I don’t think it’s brainwashing. Would you really not feel an intuition to think that something weird or spooky happened if the above example mentioned in my post happened in front of your eyes? @EricDuminil
    – user62907
    Commented Mar 17, 2023 at 7:25
  • 2
    "1. When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong. 2. The only way of discovering the limits of the possible is to venture a little way past them into the impossible. 3. Any sufficiently advanced technology is indistinguishable from magic." en.wikipedia.org/wiki/Clarke%27s_three_laws Commented Mar 17, 2023 at 16:11

13 Answers 13


Suppose for example that a person is standing on stage and says “God, if you exist, strike me with lightning right now” and a lightning strike occurs that barely misses him, is this evidence of God? Can it make it more likely that God exists?

A Bayesian analysis can be used to determine whether this evidence means we should increase our degree of belief in the existence of God. Let M1 be a model of the universe where God exists, and M2 be a model of the universe where God does not exist. The Bayes factor is the ratio of the marginal probability (integrating out an model parameters) of the observations (D) under both models:

K = P(D|M1)/P(D|M2)

Where P(D|M1) is the probability of observing a lightning strike in these circumstances is M1 is true. Using Bayes rule, we can write that as

K = P(M1|D)P(M2)/P(M2|D)P(M1)

where P(M1) represents our prior belief (before witnessing the lightning strike) that M1 is the true model (the P(D) terms cancel). Equating and rearranging,

P(M1|D)/P(M2|D) = P(D|M1)/P(D|M2) x P(M1)/P(M2)

So we can see that the Bayes factor tells us by how much the evidence changes our relative prior degree of belief in the two models to give us our posterior beliefs (the beliefs after seeing the evidence).

So if the probability of a lightning strike near the person is higher if God exists, P(D|M1), than if god does not exist, P(D|M2), then the observation will increase our rational Bayesian belief in the existence of God as K > 1.

The good thing about this is that we can steer clear of involving science, which is often used as a synonym for "rational" and gives a false impression of rationality when applied to subjects outside it's domain (e.g. anything non-falsifiable if you are a Popperian). This is not a criticism of science (I am a scientist of sorts myself), science gains value from its specificity.

The other good thing about this approach is that it separates out your prior beliefs and makes you state them explicitly (if you want to evaluate the posterior ratio) and it makes you explain how likely the observations are under your competing models.

This is how you should "...update your credence in a deity based on an event like this...". But there isn't enough information in the question to progress the analysis further at the current time.

Note if you are certain that God does not exist, the P(M1)/P(M2) is zero, and no amount of evidence will change your belief.

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    This is the best answer here by far.
    – user46575
    Commented Mar 17, 2023 at 20:01
  • 3
    Spot on, but maybe point out that "How much evidence is it?" and "How likely is it now?" are critical followup questions. Suppose I flip 1 million coins, and I propose that all of them came up heads. I check the first coin and see that it is indeed heads. The math above allows us to correctly conclude that this is evidence that all of them came up heads. It also gives us the answers to the followup questions: it's twice as likely, and the odds are now 1 in 10^(301,030)
    – Ray
    Commented Mar 17, 2023 at 20:20
  • The only reason religion has been deemed "outside the domain" of science, is because religious claims lack even the most basic of evidence. The only way anyone can believe any of it, is if they take the epistemology that they use for every other part of life, and throw that out of the window, and create a new epistemology just for religion (which means that epistemology has also not in any way proven itself to be a reliable means of discovering truth).
    – NotThatGuy
    Commented Mar 18, 2023 at 4:54
  • Beliefs about facts of reality are a poor target for concrete probability analysis (at least non-probabilistic facts - a coin flip is probabilistic). It's practically unheard of for scientists to use Bayesian analysis to argue for a scientific theory, for example. We just have no idea what the exact numeric probabilities are. This is especially problematic for unfalsifiable claims, because the probability of your claim can only ever increase, which should clearly be absurd.
    – NotThatGuy
    Commented Mar 18, 2023 at 5:09
  • I think there is some truth to Bayesian analysis in the sense of incorporating background evidence but there are also some flaws in that it pretends to have actual probabilities @NotThatGuy Prior evidence matters. If we had prior evidence of God existing, it would be “better evidence” for god to have done the coincidence in my OP. Having an actual prior PROBABILITY may seem nonsensical here but the gist is to incorporate prior information. Personally, I think your credence shouldn’t update for claims with zero prior evidence until there is “enough” evidence but that threshold is subjective.
    – user62907
    Commented Mar 18, 2023 at 5:55

No it doesn't imply a god exists. Think about it scientifically. How do physicists go about developing hypotheses and testing them? If you wanted to apply scientific reasoning, you would need to identify a plausible hypothesis and then look for, or create through experiment, evidence that could only be accounted for by that hypothesis. If you found sufficient evidence to have a high degree of statistical significance, and no evidence that undermined your hypothesis, then you can start to consider your hypothesis to have some degree of validity.

By a plausible hypothesis, I mean one that chimes with the rest of our understanding of reality. I could, for example, hypothesise that there is no Higgs boson, and that all the results that have been found at Cern have been artificially created by a kindly god who hated the idea of Peter Higgs being disappointed by the failure to find the boson that bears his name. My hypothesis would be consistent with the results but it would be utterly implausible for other reasons. Likewise with your lightening strike.

You cannot build a meaningful theory around inadequate evidence.

If I tell you that I am thinking of a number, it might be any number. If I tell you that one of its digits is 7, does that make it more likely that I am thinking of pi, for example?

Tutankhamun was male. If I find a piece of bone in a grave in Birkenhead and the bone is identified as being the bone of a male, does that make it more likely that Tutankhamun was buried in Birkenhead?

If I have a computer file and the nth bit is a zero, does that make it more likely that the file is a jpeg image of Donald Trump in a vat of strawberry jam?

The point of the ridiculous examples I have listed above is that a single event or datapoint is rarely meaningful.


If I experience a coincidence or a coincidence happens in the world that seems to be at extremely low odds, does this imply that God exists?

No. Why would it? To make such an assumption would be a non-sequitur. We have no evidence that it is possible for a god to exist, so why would we ever call upon a god as an answer to such a question?

If it doesn’t imply that God exists, can it at least make it more likely that God exists?

No. The occurrence of an event explainable by natural phenomena in no way provides any reliable information as to the existence of a being for which we have no evidence.

  • 1
    Doesn’t this just beg an explanation for what is considered explainable then? If this person did this weekly for example and had this happen, there would be a pattern. This would seem obviously unexplainable by chance and yet theoretically, it is also possible for this to occur naturally, no?
    – user62907
    Commented Mar 16, 2023 at 7:22
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    The fact something seems unexplainable to us (other than as an incredibly unlikely event) in no way can demonstrate the existence of a being for which we have no evidence. The only way you might modify your question that I can think of is by defining God with parameters that make such a demonstration possible. But remember, unless we have evidence of such a being, it doesn't really get us anywhere. Commented Mar 16, 2023 at 7:34
  • What if one argues that this very event is evidence of that being?
    – user62907
    Commented Mar 16, 2023 at 7:43
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    Then I'd want to know how it constitutes such evidence. Commented Mar 16, 2023 at 8:10
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    You say "we have no evidence that it is possible for a god to exist" but your subsequent reasoning depends on a different proposition, i.e. "it is not possible for a god to exist."
    – user46575
    Commented Mar 17, 2023 at 20:07

Can a coincidence be evidence of a god?

It depends on what you mean by evidence. According to evidential epistemology, anything that can be construed as evidence can be used to justify a conclusion. So, are coincidences in and of themselves not a form of evidence? I know of no rule that rules coincidences, arbitrariness, and randomness out as forms of evidence. And the key is 'can'. I would say it is material fact that people regularly choose to make coincidence into evidence to support their beliefs. Consider a famous cognitive bias called illusory correlation. In it, what in actuality is coincidence is viewed as correlation. This is so common, there's an expression in English for it: "giving a sign" (mindbodygreen.com). Throughout Christian history, people have been "given signs from God".

So, really, since coincidence is regularly taken to be correlational evidence, it is not a matter of 'can'. It can and does occur regularly. Your question should be written:

Should an apparent correlation taken as evidence be reduced to coincidence to reject it as evidence of a god?

And the answer depends on who you ask. A contemporary skeptic (IEP) will argue it is VERY important, given the bias of illusory correlation to reflect heavily on what probability means, and to consider that correlation is not causation, and coincidence is not correlation. Our brains routinely fool us, and so it's best to have very thorough epistemological knowledge and method to sort out coincidence, correlation, AND causation preferably in the formalisms of statistics and probability. This sort of thinking is typical of fallibilism (IEP) and naturalistic versions of scientific practice where everything, with prima facie cause, can and should be doubted. That's exactly why science takes an observation, and asks questions like "is this a coincidence or correlation" and then sets about trying to devise tests to prove things empirically one way or another. Mere a priori introspection and rational methods are simply too likely to mislead a thinker.


Of course, if there were an omnipotent and omniscient deity, it is plausible that it might intervene in human endeavors, creating the phenomena we tend to call miracles.

So that a) if there are so many coincidences that the probability they happened by chance alone is minuscule, and b) if there is no other explanation, then Yes, such coincidences would indeed suggest the existence of a deity. (Not prove, but suggest. Because there always might be non-supernatural explanations that are simply beyond our ken, at least at the time in question.)

But I would be surprised if the conditions a) and b) obtained anywhere at any time.


It is, I believe, a widespread misconception that one should not normally expect very low probability events to happen. Nothing in probability theory implies that. Anything can happen anytime, no matter how improbable.

Another widespread misconception is, I think, that probability theory excludes determinism. You may believe in determinism or not, but there is nothing in probability theory that you could use to scientifically exclude it. I say that because no evidence, no matter how striking, can ensure existence or non-existence of anything without faith in certain principles on top of purely scientific method. Your belief or refutation of determinism, in particular, will be always independent of any scientific evidence, be it probabilistic, quantum-mechanical or any other.


Carl Jung's concept of Synchronicity, as he intended it, says that "the structure of reality includes a principle of acausal connection which manifests itself most conspicuously in the form of meaningful coincidences." (R. Main, 1997). This would be a property of the universe, so could include agency of a pantheistic god or super-intelligence.

For Christianity, however, at least according to G.K. Chesterton's Orthodoxy (1908), the innovation of Christianity is that "it divided God from the cosmos. That transcendence and distinctness of the deity ... was really the only reason why any one wanted to be a Christian." So in this case what goes on in the cosmos is separate from God. God is not responsible for the manifestations of cosmic super-intelligence and the latter is not evidence of the former.

(Chesterton does say "Briefly, it divided God from the cosmos." But this temporary status does not affect the point that god is not necessarily connected to the goings-on of the universe.)

P.S. Synchronicities are impossible to prove, even to yourself. Possibly by design.

  • If I'm reading this right, you're basically saying "[the Christian] God doesn't act through coincidence (but something else does?)". But, how I read the question, is that it's asking about events that would be highly unlikely if God didn't exist, but likely if he did. Kind of how it would be highly unlikely for my phone to slide across my desk without my involvement, but likely with my involvement. So your introduction of a second god-like entity to which you can attribute whatever you want, is technically an answer, but really sidesteps the issue rather than addressing it.
    – NotThatGuy
    Commented Mar 18, 2023 at 4:35
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    The first paragraph of your own link says that's pseudoscientific with dubious experiments involving astrology. So I think I'll keep my reservations.
    – NotThatGuy
    Commented Mar 18, 2023 at 4:40
  • @NotThatGuy I'm saying there are possible definitions of god that would separate god from supernatural agency and where the latter wouldn't be evidence of the former. Commented Mar 18, 2023 at 9:39

Does an unlikely event make the existence of a god more likely? Maybe, but that might be the wrong way to think about it.

Let's approach this scientifically.

So we have an observation: a man asks God to strike him with lightning, and lightning strikes next to him.

One can't really conclude much from observation alone (rare events happen all the time), but we do have a hypothesis:

Hypothesis: asking God to strike you with lightning makes lightning strike next to you.

Supposing that asking God doesn't always work (because "you need enough faith", perhaps), we can modify this to: asking God to strike you with lightning makes it more likely that lightning will strike next to you.

In order to test this hypothesis, we can turn this into a scientific study, where you have one group of people who ask God, and one group who doesn't. Then you record lightning strikes and apply statistics to it to determine whether to accept the hypothesis with some probability.

Now lightning strikes are presumably too rare to get statistically significant results, but you could apply the same idea to more likely events, such as for healing, which people have done (and this "failed to produce significant findings").

Confounding variables

Consider something else one might ask God for: asking God to help you find a job.

Let's say you're doing a cohort study instead of a controlled one (i.e. you just look at what happened instead of controlling anything). Now you very well may find a significant correlation between prayer and getting a job offer.

But one obvious confounding variable would be how much you want a job. If you really want a job, you may search harder and prepare better (so you're more likely to get a job offer), and you may also be more likely to pray. So any correlation you find between prayer and getting a job offer may simply be due to the fact that both are more likely when someone really wants a job.

If you consider believers who pray and non-believers, this would have the same problem. If you consider any believers and non-believers, there would still be confounding variables in terms of how employers view believers and non-believers, how wealthy the average believer or non-believer is, which resources or opportunities they might have access to (e.g. through their church), etc.

These things can be accounted for, but one should be aware of it before concluding anything.

What did we prove?

We could consider another case, which I've heard cited a few times by theists: asking God to help with addiction or depression.

In this case, one might even concede that prayer could work for this, but not because God exists. Instead, there may be some psychological benefits to putting some of one's burdens on someone else (even if they are merely imagined), from "knowing" there's someone looking out for you, from explicitly verbalising your desires, from taking some time to calm yourself, etc.

So if prayer has been proven to improve mental states, you still wouldn't be much closer to proving that God had anything to do with it.

If none of the above applies, then we might have some useful evidence.

You ask God for lightning, you get lightning (with statistical significance).

One could still say it's due to aliens with advanced technology, quantum stuff, any natural phenomenon that we're not familiar with yet, or psychic powers, and any given one of these explanations may be more or less likely, depending on the nature of the evidence. Different people would have different standards for when they'd accept it was a god instead of any of these things (or instead of simply sticking to "I don't know").

If one concludes that a god was responsible, the next obvious question would be: which god? There are multiple religions, and even within e.g. Christianity, there are vastly different beliefs of what God is like. So one would also need any evidence for a god to include details about that god. Otherwise you'd just be left with: some god exists, but I know nothing about them nor about what they want nor about what they offer, so their existence has no bearing on my life.

Theists can and do say God shall not be tested, but this would really be the only way to definitively reject the possibility of coincidence (even if it's much harder to definitively attribute things to God after rejecting coincidence).

  • I would like to point out that any good theist will tell you that God works through natural means - hence the existence of naturalistic explanations for why prayer works in some scenarios should not be taken as evidence against God's hand in it, though I concede that from a scientific standpoint it also cannot be taken as evidence for.
    – user46575
    Commented Mar 18, 2023 at 19:32
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    @DarkMalthorp For what I've written, I don't think it matters how God works, just that God works (in a reliable manner, in response to human actions such as prayer). There are also different views for how God works. The existence of naturalistic explanations would make God's involvement an unnecessary hypothesis.
    – NotThatGuy
    Commented Mar 18, 2023 at 21:54

Yes, absolutely, but evidence for a hypothesis does not mean the hypothesis is true!

In your case the evidence is extremely weak, but it is still evidence. We can know it is evidence because if you repeated the experiment 10 times and got the same result, we would not dismiss that as a coincidence, and there isn't some magic number of times you repeat an experiment where it switches from coincidence to evidence.

Consider flipping a coin three times and coming up with heads 3 times. Now consider the hypotheses:

  1. Is the coin fair?
  2. Does it have a 57% bias toward heads?
  3. Does it have a 58% bias towards heads?
  4. Does it have a 59% bias towards heads?
  5. Does it have only two tails sides?

Is the result evidence for hypothesis #3? Yes! But that result was evidence for all of 1-4. If someone was constantly reiterating that this is evidence for #3, but never mentioning the rest, I would wonder whether they had some strange agenda. Now consider the literally billions of hypotheses that also fit the lighting event you describe. Why did you decide to focus on a very specific version of one with a god?


Can a coincidence be evidence of a god?

We feel that things also happen in life that are not coincidences. And the term ‘god’ has different meaning in different religions. So I would say, “No, this is never an evidence of existence of a god. But this is a clear evidence of consciousness.”

Something that someone forgot also happens in time and often reminds us that behind all these happenings there is a conscious that 'works' without failure. And among these happenings there may be bad as well as good. Since we feel that bad things also happens, I wish to substitute the term ‘god’ by Consciousness. So there is no need to carry a devil with us throughout our life :)

I didn’t mean that this is an evidence of all the attributes mentioned in the description. But that also is possible to discover them. For that, misconceptions need to be uncovered.


Perhaps if you line up a billion people, half of them say the magic words, and more lightning hits those people than hits the control people, we have evidence to believe something.

Even then "God exists" is a vague statement. There are many different gods that are claimed to exist. The most we could conclude is that these particular words make it more likely to be struck by lighting. Anything more general requires further testing.


(Note: I will assume by "God" you mean the monotheistic/platonist understanding of God as the uncreated first principle by which all things are created.)

This question, on some level, doesn't make sense. (But on another level, it does.)

What do you mean by it being "likely" that God exists? He either exists or he doesn't. If he exists, then he exists absolutely, in all universes, even in all possible universes. Godel, in his famous proof for God's existence, called this property "necessary existence". It is also the standard Christian interpretation for the meaning of the name of God as revealed in Exodus 3. Platonists have a similar view of God as the necessary first principle.

Thus, according to the monotheist/Platonist view, God cannot not exist. It's not important to go in detail on Godel's full proof of God's existence, only to understand that God, by definition, necessarily exists, and thus the only way for him to not exist is if it isn't possible for him to exist. After all, we cannot define logical inconsistencies and force them to be true by declaring "necessary existence". However, if something is possible to exist and has the property of necessary existence by definition, then it must exist.

Therefore there is no probability distribution that we can sample from which contains both "God exists" and "God does not exist". If it is possible for God to exist, then he exists necessarily and irrevocably in all possible worlds.

That is why, from a strictly logical perspective, evidence for or against God is never satisfying - in some sense God is above evidence, even if he doesn't exist. By definition, he precedes all evidence, and if he does exist, he supersedes and overrides all evidence. If you are to answer the question "does God exist?" your answer must - from a strictly logical standpoint - be an axiom and is therefore unsusceptible to evidence, whether that answer is "yes" or "no".

With all that being said, what are we to do? How can we know whether God exists or not?

Well it ultimately does come down to evidence - but not in a logical way, see above.

A coincidence is not logical (or even scientific) evidence for God, but it may make the "God is impossible" perspective less plausible. The question is, what axiom better matches the world I observe? God exists, or God doesn't exist? Those who disbelieve may do so because the idea of God seems contrary to their own understanding of the world. Those who believe do so because God's non-existence would seem contrary to their understanding of the world. In the words of C.S. Lewis, "I believe in Christianity as I believe that the sun has risen, not only because I see it but because by it, I see everything else." This statement rings true to me also: My belief in Christ is foundational to how I see the world, and is such a powerful organizing and illuminating principle that I would be blind without it.

So, if you want to find out whether Christ is the real God: I recommend you reflect on what Christianity teaches (I can recommend crossway, desiringgod, or gotquestions.org for an introduction), and also talk to your Christian friends, maybe try going to church or reading the Bible. Does Christianity seem to fill a void in your worldview, seem to accord with and help you understand your own experiences (including coincidences - but also everything else)? If yes, then you should consider the possibility it is true.

If you are curious about other faiths, Islam, Judaism, Hinduism, or whatever, I can't help you with any specifics - I'm a Christian after all - but I think there's some clear analogous steps you can take for other religions to see if they make sense. Talk to practitioners, learn about and reflect on their doctrines and rituals.

  • "It's not important to go in detail on Godel's full proof, only to understand that God, by definition, necessarily exists" - uhhh... the proof of necessary existence seems rather important to the claim of necessarily existence. Otherwise you just have a baseless claim. In any case, Godel is just begging the question by inserting existence into the definition of God in order to prove God's existence (replace "God" with "the Flying Spaghetti Monster" and it works equally well).
    – NotThatGuy
    Commented Mar 18, 2023 at 3:48
  • If something has any sort of physical presence, or interacts with the physical world in any way, that would be evidence. So for God to be "above evidence", this would mean they do not interact with the physical world in any way, and therefore their existence is irrelevant to us. Also, we do not have access to objective truth, so essentially everything we know about reality would be derived from some sort of evidence, and this is based on likelihoods (if you see an apple in the distance, you may not know whether it's real or a picture, and getting a better look changes the likelihoods of each).
    – NotThatGuy
    Commented Mar 18, 2023 at 3:59
  • I think you missed my point. Godel's proof is not a proof of necessary existence; God by definition necessarily exists. Therefore, God either exists or cannot possibly exist.
    – user46575
    Commented Mar 18, 2023 at 19:20
  • In response to the 2nd comment - you make an assumption of materialism; if human beings are spiritual as well as material, then even if God never interacts with the physical world in any way, his existence may still be relevant to us. However, I and other monotheists do believe God interacts with the physical universe in all things that happen, as all things happen according to his will and design, but I doubt you would consider every event to be evidence for God.
    – user46575
    Commented Mar 18, 2023 at 19:25
  • actually, you have pointed out that my paragraph on necessary existence does not make a clear statement of the claim I was thinking of... Allow me to rephrase.
    – user46575
    Commented Mar 18, 2023 at 19:34

TL;DR: Yes. It follows from Bayes' theorem.


We are going to consider a total of 5 events, whose probabilities and conditional probabilities are of interest:

First there are the three events that we are interested in:

  • (Event A) God exists and is willing to strike you with lightning if you ask;
  • (Event B) Some other confounding factor increases your chances of being hit by lightning when you ask;
  • (Event C) Lightning is entirely random and not influenced by you asking for it.

Now you ask God to strike you with lightning, and we have two more events:

  • (Event L) You are struck by lightning;
  • (Event ~L) You are not struck by lightning.

Rephrasing the question

Your question can now be rephrased as:

Is P(A | L) > P(A)?

P(A | L), which you can pronounce "Probability of A knowing L", is the conditional probability of event A after we've observed event L. We want to know whether event A is "more likely after having observed L", and this is what the number P(A | L) is measuring.

A few observations about events A and B

Note that event B is very important and we shouldn't forget about it in our analysis. It includes possible confounding factors such as "Someone has a weather-influencing machine and is using it to make you believe in God", but also "In your search for God, you stood in the middle of a field during a storm and raised your arms to talk to God".

Also note that event A is not just "God exists" but "God exists and is willing to strike you with lightning when you ask". Nevertheless, if P(A|L)

Prior probabilities

Before we apply Bayes' theorem, we need to have prior values for the probabilities of the events we're interested in. These are going to be a bit arbitrary, but as long as they are nonzeroes, the exact values don't really matter since the question is not to find an exact value for P(A|L), but only to find whether P(A|L) is greater than P(A) or not. We could perform the analysis completely abstractly without filling in exact values, but it would be a lot more tedious, so let's choose a few values for simplicity:

  • P(A) = 0.0000001% (it seems extremely unlikely that God would strike you on demand)
  • P(B) = 1% (it's possible that in your attempt to talk to God, you take risks and increase your own probability of being hit by lightning)
  • P(C) = 90% (events A and B are not necessarily disjoint, so the logical value for P(C) must be somewhere between 89.99% and 90%; I'm rounding up for simplicity)
  • P(L | C) = 0.0000003% (I googled "probability of being hit by lightning" and the first result said it was one in a million for a whole year, and I divided that by 360 to get the probability for a day)
  • P(L | A) = 90% (arbitrary)
  • P(L | B) = 5% (arbitrary)

Note that I chose P(L | A) and P(L | B) rather arbitrarily. But as I said, the exact numbers don't really matter, they just make it easier to reason.

Bayes' theorem

Bayes' theorem states:

P(A | L) = P(L | A) P(A) / P(L)

P(A | L) = P(L | A) P(A) / (P(L | A) P(A) + P(L | B) P(B) + P(L | C) P(C))

With our values, we get:

P(A | L) = 0.9 x 0.000000001 / (0.9 x 0.000000001 + 0.05 x 0.01 + 0.000000003 x 0.9)

P(A | L) = 0.00018%


P(A) = 0.0000001%, and P(A | L) = 0.00018%.

Unsurprisingly, you were correct. The probability that God exists and is willing to strike you with lightning on demand has been bumped up by a factor of about 2000 when you were hit by lightning!

The probability is still pretty low, because of the possibility of confounding factors, which we included as event B. If you redo the same calculations but with P(B) = 0 (meaning there are no possible factors that would explain the lightning, except a pure coincidence or God's intervention), then you would find a much higher value for P(A | L) after being hit by lightning.

Further discussion

You might be disappointed by the use of arbitrary values in my analysis. Those arbitrary values are actually not necessary. You could redo the calculations, leaving P(A), P(L | A), P(B), P(L | B) without substituting numeric values for these four numbers. Then, a mathematical analysis of the resulting formula will show that if P(L | A) > P(L | C), then P(A | L) > P(A | ~L), and thus P(A | L) > P(A).

Since this is philosophy.stackexchange and not math.stackexchange or stats.stackexchange, I chose to use numeric values for simplicity.

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