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The statistics of miracles
delirium happy
rho
I've been pondering the statistics of miracles and the paranormal recently. Now, I generally tend to be cynical and suspicious about things of that nature, but I also try to keep an open mind. I think that such things are possible, but that most purported instances aren't genuine. Or, to put it another way, I take as the null hypothesis the credible and the mundane. And I do wonder about the statistical validity of some claims.

A very brief lesson in statistics for those who know nothing it: imagine that you have a brand new die. You roll it three times and score three consecutive 6s. At this point you might well think "hang on a second, there's only a 1 in 216 chance of getting three consecutive 6s with a fair die. This one must be fixed!" If you did think that though, then you'd be wrong. Human intuition and statistics really don't allign with each other very well. If I'd said that you took 216 dice and threw each of them thrice, then it's no surprise at all if one of them landed as a 6 every time. And while very few people roll 200 different dice with any great frequency, we generally do perform over 200 actions each day, each of which might have one possible outcome with a probability of around 1 in 200. The problem is, the human brain doesn't pick up on normal things. If things happen exactly as expected, we often don't even stop to think that there was a possibility of it happening any other way. As you refreshed your friends page, you probably didn't consider that you could have accidentally missed the refresh button.

The way that statistics tends to combat this tendency to only notice the incredible and never the mundane is to say that only observations made after a hypothesis count. So, first we make preliminary observations, and come up with a vague idea that we might know what is going on. Then we codify this belief into a definite hypothesis that we can test. We then make further new observations to try to test this hypothesis, which allows us to see whether something was a real effect, or just something that we picked up due to our tendency to only notice the unlikely. So, to go back to our dice example, having rolled our three 6s, we then make the hypothesis that it's fixed (we might say that it will always give a 6, or we might say that it has a chance higher than 1 in 6 of givign a 6, or whatever) and we would then make subsequent rolls, to see if the hypothesis was true. So, we might decide to roll the die 100 more times. If we get a 6 every time, then we have a real effect. If we only get sixteen 6s then we see that it wasn't a real effect, but a random chance that we picked up on due to our tendency to notice the unusual.

Now, lesson in statistics over, we return to the point. With miraculous or paranormal events, it's often not possible to perform any sort of rigorous statistical analysis. If there's a one-off miraculous-seeming event, then it simply isn't possible to perform the repeat tests. Even with things that are repeated, such as miraculous healings, we simply don't have the full information. Sure, we may hear about the people who are terminally ill, pray and get better, but we don't hear so much about the people in the same people who pray and then die, or the people who don't pray but still get better. After all, even million to one chances are likely to happen occasionally with a global population of over six billion.

So how do we judge whether this sort of thing is significant or not? If we get truly spooked by thinking the exact same thing as someone else, how do we know if we were actually being telepathic or if it was just a random coincidence? I'm not entirely sure, though I do feel certain that smart people must have thought about this sort of thing a lot and given some good answers, only I don't know what they are. What I'm thinking though, is that we need to try to estimate the probabilities of these so called miraculous events.

Consider, we only need to perform 50 actions a day and live for 60 years to get over a million actions in our life. Given that, we really shouldn't be all that surprised if we have something happening to us with million to one odds. And if we consider all the people who have ever lived through the entire of recrded human history, then we start getting into the realm of numbers bigger than I can comprehend, and we really have to expect that some staggeringly unlikely events will actually have happened.

The problem is, figuring out just how unlikely something actually is. Turning water into wine or Ford Prefect into a penguin are both staggeringly unlikely. Theoretically possible, when you consider things in terms of quantum, but massively, massively unlikely. The probability of a single molecule of water (or probably three molecules) randomly transmuting into a single molecule of ethanol is extremely and uncalculably small. In a litre of water there are over a million million million million molecules. We're talking very high improbabilities here.

But on the other hand, consider someone thinking of a loved one dying just at the moment that they are in a life threatening situation. That really isn't that unlikely. There are only a finite number of things a person can think about, and they have to be thinking about something at the point where the loved on is in trouble. It isn't likely that you'll get that particular thought purely by coincidence, but it isn't so unlikely that we can chalk it up as only being possible due to a miracle or the paranormal.

But just how many miracles should we expect as our background level? An average person probably has a lifespan of around about thirty to forty million minutes. If we consider that it might take a minute for something to happen, then that would mean an expected value of about thirty to forty million to one shots happening in a lifetime. If you consider that you might know about a hundred people, then you should expect that either you or someone you know should have a billion to one miracle at some point in their life. And worldwide, you could expect a hundred million billion to one miracle during you rlifetime.

Of course, this is an oversimplification. Qunatising things into minutes doesn't really reflect the real world. Miraculous things are much more likely to happen (or be observed) at some points than others: not many people are going to observe miracles when they're asleep. Some things that are fantastically unlikely may also be fairly mundane in appearance and may go unnoticed. And so on and so forth. It does give something of a good indication as to where I'm coming from though.

So what we really have to consider, when trying to determine whether there really is something sinister/divine/inexplicable/whatever going on, is just how often do these minor miracles occur. If they're much more frequent than our expected value (which can be only one instance for the extremely improbable) then it's reasonable to believe that they truly are more than just a random chance, otherwise, there's no reason to suspect anything more.

Of course, this doesn't really help all that much, because trying to predict that sort of probability is nigh on impossible. It would need a miracle.

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As you refreshed your friends page, you probably didn't consider that you could have accidentally missed the refresh button. oh yes i do. with the frequency that I hit f6 (tile all tabs) instead of f5 (refresh), the whole irc channel i'm in hears a rant about it every single time.

(yes, i did read the rest and found it interesting, i just harp on small things and only comment about those. for the record i'm of the contingent that a spiders web is a miracle because i couldn't make one and this outlook keeps me happy.)

Oooh, that made for some interesting thinking! Thank you for that one :-D

I'm a cynic, but I have gotten better reliability from prayer and demon summoning than from science labs. My conservation labs never worked. My coffee cup calorimeter experiment, where you measure heat loss over time - gained energy. Whereas I generally got the things I prayed for or summoned up a demon to give me. And I generally asked for things I didn't think I could possibly get. Just one of those weird things... but I still believe in the laws of conservation.

Miracles and statistics

(Anonymous)
I often wondered, if out of a hundred miracles performed at Lourdes from x million visitors, the same hundred miracles have been quietly performed (Unreported) from a count of the same number of people (x million) elsewhere in the world.
I have a feeling miracles in the grandest sense - from unredeemable illness to complete cures - are a continuous affair all over the globe. I'm guessing there's a web site somewhere where people can report miracles, and if so I'm also guessing it's tainted with the y million people that don't believe oe elevate such things.

Tony Troy



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