# Prayer and Probability

The reader might wonder about the relation between the previous post and my discussion of Arman Razaali. If I could say it is more likely that he was lying than that the thing happened as stated, why shouldn’t they believe the same about my personal account?

In the first place there is a question of context. I deliberately took Razaali’s account randomly from the internet without knowing anything about him. Similarly if someone randomly passes through and reads the previous post without having ready anything else on this blog, it would not be unreasonable for them to think I might have just made it up. But if someone has read more here they probably have a better estimate of my character. (If you have read more and still think I made it up, well, you are a very poor judge of character and there is not much I can do about that.)

Second, I did not say he was lying. I said it was more likely than the extreme alternative hypothesis that the thing happened exactly as stated and that it happened purely by chance. And given later events (namely his comment here), I do not think he was lying at all.

Third, the probabilities are very different.

## “Calculating” the probability

What is the probability of the events I described happening purely by chance? The first thing to determine is what we are counting when we say that something has a chance of 1/X, whatever X is. Out of X cases, the thing should happen about once. In the Razaali case, ‘X’ would be something like “shuffling a deck of cards for 30 minutes and ending up with the deck in the original order.” That should happen about once, if you shuffle and check your deck of cards about 10^67 times.

It is not so easy to say what you are counting if you are trying to determine the probability of a coincidence. And one factor that makes this feel weirder and less probable is that since a coincidence involves several different things happening, you tend to think about it as though there were an extra difficulty in each and every one of the things needing to happen. But in reality you should take one of them as a fixed fact and simply ask about the probability of the other given the fixed thing. To illustrate this, consider the “birthday problem“: in a group of 23 people, the chance that two of them will have the same birthday is over 50%. This “feels” too high; most people would guess that the chance would be lower. But even without doing the math, one can begin to see why this is so by thinking through a few steps of the problem. 22 days is about 6% of the days in a year; so if we take one person, who has a birthday on some day or other, there will be about a 6% chance that one of the other people have the same birthday. If none of them do, take the second person; the chance one of the remaining 21 people will have the same birthday as them will still be pretty close to 6%, which gets us up to almost 12% (it doesn’t quite add up in exactly this way, but it’s close). And we still have a lot more combinations to check. So you can already start to see that how easy it will turn out to be to get up to 50%. In any case, the basic point is that the “coincidence” is not important; each person has a birthday, and we can treat that day as fixed while we compare it to all the others.

In the same way, if you are asking about the probability that someone prays for a thing, and then that thing happens (by chance), you don’t need to consider the prayer as some extra factor — it is enough to ask how often the thing in question happens, and that will tell you your chance. If someone is looking for a job and prays a novena for this intention, and receives a job offer immediately afterwards, the chance will be something like “how often a person looking for a job receives a job offer.” For example, if it takes five months on average to get a job when you are looking, the probability of receiving an offer on a random day should be about 1/150; so out 150 people praying novenas for a job while engaged in a job search, about 1 of them should get an offer immediately afterwards.

What would have counted as “the thing happening” in the personal situation described in the last post? There are a number of subjective factors here, and depending on how one looks at it, especially depending on the detail with which the situation is described. For example, as I said in the last post, it is normal to think of the “answer” to novena on the last day or the day after — so if a person praying for a job receives an offer on either of those days, they will likely consider it just as much of an answer. This means the estimate of 1/150 is really too low; it should really be 1/75. And given that many people would stretch out the period (in which they would count the result as an answer) to as much as a week, we could make the odds as high as 1/21. Looking loosely at other details could similarly improve the odds; e.g. if receiving an interview invitation that later leads to a job is included, the odds would be even higher.

But since we are considering whether the odds might be as bad as 1/10^67, let’s assume we include a fair amount of detail. What are the odds that on a specific day a stranger tells someone that “Our Lady wants you to become a religious and she is afraid that you are going astray,” or words to that effect?

The odds here should be just as objective as the odds with the cards — there should be a real number here — for reasons explained elsewhere, but unfortunately unlike the cards, we have nowhere near enough experience to get a precise number. Nonetheless it is easy to see that various details about the situation made it actually more likely than it would be for a perfectly random person. Since I had a certain opinion of my friend’s situation, that makes it far more likely than chance that other people aware of the situation would have a similar opinion. And although we are talking about a “stranger” here, that stranger was known to a third party that knew my friend, and we have no way of knowing what, if anything, might have passed through that channel.

If we arbitrarily assume that one in a million people in similar situations (i.e. where other people have similar opinions about them) hear such a thing at some point in their lives, and assume that we need to hit one particular day out of 50 years here, then we can “calculate” the chance: 1 / (365 * 50 * 1,000,000), or about 1 in 18 billion. To put it in counting terms, 1 in 18 billion novenas like this will result in the thing happening by chance.

Now it may be that one in a million persons is too high (although if anything it may also be too low; the true value may be more like 1 / 100,000, making the overall probability 1 / 180 million). But it is easy to see that there is no reasonable way that you can say this is as unlikely as shuffling a deck of cards and getting it in the original order.

## The Alternative Hypothesis

A thing that happens once in 18 billion person days is not so rare that you would expect such things to never occur (although you would expect them to most likely not happen to you). Nonetheless, you might want to consider whether there is some better explanation than chance.

But a problem arises immediately: it is not clear that the alternative makes it much more likely. After all, I was very surprised by these events when they happened, even though at the time I did attribute an explicitly religious explanation. Indeed, Fr. Joseph Bolin argues that you should not expect prayer to increase the chances of any event. But if this is the case, then the odds of it happening will be the same given the religious explanation as given the chance explanation. Which means the event would not even be evidence for the religious explanation.

In actual fact, it is evidence for the religious explanation, but only because Fr. Joseph’s account is not necessarily true. It could be true that when one prays for something sufficiently rare, the chance of it happening increases by a factor of 1,000; the cases would still be so rare that people would not be likely to discover this fact.

Nonetheless, the evidence is much weaker than a probability of 1 in 18 billion would suggest, namely because the alternative hypothesis does not prevent the events from remaining very unlikely. This is an application of the discussion here, where I argued that “anomalous” evidence should not change your opinion much about anything. This is actually something the debunkers get right, even if they are mistaken about other things.

# Might People on the Internet Sometimes Tell the Truth?

## Lies and Scott Alexander

Scott Alexander wrote a very good post called Might People on the Internet Sometimes Lie, which I have linked to several times in the past. In the first linked post (Lies, Religion, and Miscalibrated Priors), I answered Scott’s question (why it is hard to believe that people are lying even when they probably are), but also pointed out that “either they are lying or the thing actually happened in such and such a specific way” is a false dichotomy in any case.

In the example in my post, I spoke about Arman Razaali and his claim that he shuffled a deck of cards for 30 minutes and ended up with the deck in its original order. As I stated in the post,

People demonstrably lie at far higher rates than 1 in 10^67 or 1 in 10^40. This will remain the case even if you ask about the rate of “apparently unmotivated flat out lying for no reason.” Consequently, “he’s lying, period,” is far more likely than “the story is true, and happened by pure chance.” Nor can we fix this by pointing to the fact that an extraordinary claim is a kind of extraordinary evidence

But as I also stated there, those are not the only options. As it turns out, although my readers may have missed this, Razaali himself stumbled upon my post somewhat later and posted something in the comments there:

At first, I must say that I was a bit flustered when I saw this post come up when I was checking what would happen when I googled myself. But it’s an excellent read, exceptionally done with excellent analysis. Although I feel the natural urge to be offended by this, I’m really not. Your message is very clear, and it articulates the inner workings of the human mind very well, and in fact, I found that I would completely agree. Having lost access to that Quora account a month or two ago, I can’t look back at what I wrote. I can easily see how the answer gave on Quora could very easily be seen as a lie, and if I read it with no context, I would probably think it was fake too. But having been there at the moment as I counted the cards, I am biased towards believing what I saw, even though I could have miscounted horrendously.

Does this sound like something written by one of Scott Alexander’s “annoying trolls”?

Not to me, anyway. I am aware that I am also disinclined for moral reasons to believe that Razaali was lying, for the reasons I stated in that post. Nonetheless, it seems fair to say that this comment fits better with some intermediate hypothesis (e.g. “it was mostly in order and he was mistaken”) rather than with the idea that “he was lying.”

## Religion vs. UFOs

I participated in this exchange on Twitter:

Ross Douthat:

Of what use are our professionally-eccentric, no-heresy-too-wild reasoners like @robinhanson if they assume a priori that “spirits or creatures from other dimensions” are an inherently crazy idea?: https://overcomingbias.com/2021/05/ufos-say-govt-competence-is-either-surprisingly-high-or-surprisingly-low.html

But we don’t want to present ourselves as finding any strange story as equally likely. Yes, we are willing to consider most anything, at least from a good source, & we disagree with others on which stories seem more plausible. But we present ourselves as having standards! 🙂

Me:

I think @DouthatNYT intended to hint that many religious experiences offer arguments for religions that are at least as strong as arguments from UFOs for aliens, and probably stronger.

I agree with him and find both unconvincing.

But find it very impressive you were willing to express those opinions.

You can find videos on best recent evidence for ghosts, which to me looks much less persuasive than versions for UFOs. But evidence for non-ghost spirits, they don’t even bother to make videos for that, as there’s almost nothing.

Me:

It is just not true that there is “almost nothing.” E.g. see the discussion in my post here:

Miracles and Multiple Witnesses

Robin does not respond. Possibly he just does not want to spend more time on the matter. But I think there is also something else going on; engaging with this would suggest to people that he does not “have standards.” It is bad enough for his reputation if he talks about UFOs; it would be much worse if he engaged in a discussion about rosaries turning to gold, which sounds silly to most Catholics, let alone to non-Catholic Christians, people of other religions, and non-religious people.

But I meant what I said in that post, when I said, “these reports should be taken seriously.” Contrary to the debunkers, there is nothing silly about something being reported by thousands of people. It is possible that every one of those reports is a lie or a mistake. Likely, even. But I will not assume that this is the case when no one has even bothered to check.

Scott Alexander is probably one of the best bloggers writing today, and one of the most honest, to the degree that his approach to religious experiences is somewhat better. For example, although I was unfortunately unable to find the text just now, possibly because it was in a comment (and some of those threads have thousands of comments) and not in a post, he once spoke about the Miracle of the Sun at Fatima, and jokingly called it something like, “a glitch in the matrix.” The implication was that (1) he does not believe in the religious explanation, but nonetheless (2) the typical “debunkings” are just not very plausible. I agree with this. There are some hints that there might be a natural explanation, but the suggestions are fairly stretched compared to the facts.

## December 24th, 2010 – Jan 4th, 2011

What follows is a description of events that happened to me personally in the period named. They are facts. They are not lies. There is no distortion, not due to human memory failures or anything else. The account here is based on detailed records that I made at the time, which I still possess, and which I just reviewed today to ensure that there would be no mistake.

At that time I was a practicing Catholic. On December 24th, 2010, I started a novena to Mary. I was concerned about a friend’s vocation; I believed that they were called to religious life; they had thought the same for a long time but were beginning to change their mind. The intention of the novena was to respond to this situation.

I did not mention this novena to anyone at the time, or to anyone at all before the events described here.

The last day of the novena was January 1st, 2011, a Marian feast day. (It is a typical practice to end a novena on a feast day of saint to whom the novena is directed.)

On January 4th, 2011, I had a conversation with the same friend. I made no mention at any point during this conversation of the above novena, and there is no way that they could have known about it, or at any rate no way that our debunking friends would consider “ordinary.”

They told me about events that happened to them on January 2nd, 2011.

Note that these events were second hand for me (narrated by my friend) and third hand for any readers this blog might have. This does not matter, however; since my friend had no idea about the novena, even if they were completely making it up (which I believe in no way), it would be nearly as surprising.

When praying a novena, it is typical to expect the “answer to the prayer” on the last day or on the day after, as in an example online:

The Benedictine nuns of St Cecilia’s Abbey on the Isle of Wight (http://www.stceciliasabbey.org.uk) recently started a novena to Fr Doyle with the specific intention of finding some Irish vocations. Anybody with even a passing awareness of the Catholic Church in Ireland is aware that there is a deep vocations crisis. Well, the day after the novena ended, a young Irish lady in her 20’s arrived for a visit at the convent. Today, the Feast of the Immaculate Conception, she will start her time as a postulant at St Cecilia’s Abbey.

Some might dismiss this as coincidence. Those with faith will see it in a different light. Readers can make up their own minds.

January 2nd, 2011, was the day after my novena ended, and the day to which my friend (unaware of the novena) attributed the following event:

They happened to meet with another person, one who was basically a stranger to them, but met through a mutual acquaintance (mutual to my friend and the stranger; unknown to me.) This person (the stranger) asked my friend to pray with her. She then told my friend that “Our Lady knows that you suffer a lot… She wants you to become a religious and she is afraid that you are going astray…”

Apart from a grammatical change for context, the above sentences are a direct quotation from my friend’s account. Note the relationship with the text I placed in bold earlier.

## To be Continued

I may have more to say about these events, but for now I want to say two things:

(1) These events actually happened. The attitude of the debunkers is that if anything “extraordinary” ever happens, it is at best a psychological experience, not a question of the facts. This is just false, and this is what I referred to when I mentioned their second error in the previous post.

(2) I do not accept a religious explanation of these events (at any rate not in any sense that would imply that a body of religious doctrine is true as a whole.)