Friday, December 05, 2014

Fermi Problem Friday: Quantifying Miracles

Lazarus rising from the dead,
as did Janina Kolkiewicz and Walter Williams earlier this year.
It's a rare event, but is it miraculous?
The holiday season is upon us, and we will soon be inundated with stories of miracles from Frosty the Snowman to 34th Street and religious celebrations of one kind or another.

Talk of miracles, of course, is not limited to the holidays. A quick search of the the news shows that the word comes up with amazing frequency, often used in connection with medical stories, and shockingly often by doctors.

I usually think of doctors as applied scientists, so it seems strange to me that they would use the word "miracle" so often. There are certainly other words that they could choose when something unusual happens. Words like anomaly, stroke of luck,lucky break, or beat the odds. But when you enter these words in Google Trends, which can show you the frequency of words searched on the internet, they all lose out to "miracle" in popularity.

So I started wondering, what does it take for something to be "miraculous" in the way the word is often used?

The story of a recent "miracle birth" provides one estimate. A couple expecting a child were trapped in their house by a snow storm in Buffalo, NY when the wife went into labor. The husband stepped outside only to meet an ambulance driver who turned out to be a maternity nurse, and when they finally headed to the hospital, along the way they met another maternity nurse stuck in the snow. The two nurses successfully delivered the baby despite the challenging weather. The father in the story claimed it had to be a miracle to happen across not one, but two, maternity nurses right when they were most needed, saying, “It’s not odds; it’s God.”

But what were the odds, really?

In the US, there are about three million professional nurses. Of those, let's guess that one in fifty are maternity experts. So there are about 60,000 of them in the US. If you live in the States, the chance that the next person you meet will be a maternity nurse is about 6000/400,000,000, because there are about 400 million people in the US. So there's about a one in 6,666 chance the next person you bump into will be a maternity nurse.

The chance that you would meet two maternity nurses in a row (assuming you're not actually in a maternity ward, that is) is about (1/6,666)x(1/6,666) = 1/44,435,556.

So if it was odds after all, and not supernatural intervention, a "miracle" is really just a one in 44 million event.

People who win the lottery often thank God for their good fortune. The odds against winning are pretty high, typically about one in 13 million in many state lotteries. But if ten million people play a lottery with those odds, there's very good chance that someone will win almost every time there's a drawing, so it's clearly not a miracle to anyone except the winner. But for the lucky winner who feels like the beneficiary of a supreme being's good will, that would suggest "miracles" can be one in 10 million events as well.

Mathematician John Littlewood attempted an armchair analysis of miracles. But instead of looking at the miracles people claim to experience, he decided to come up with a definition of miracles. Specifically, Littlewood supposed a one-in-a million event would seem pretty miraculous to most people.

He also assumed that a person experiences one detectable event roughly every second, where an event can be anything from hearing a sound to seeing an object to running into a maternity nurse. By that definition, you detect sixty events every waking minute and 3600 events an hour. For some reason, Littlewood only assumed an eight hour period per day for someone to rack up noticeable events, or about 28,800 evetns a day, which leads to a grand total approaching a million events a month. (People are usually awake more like sixteen hours a day, so I would claim that you experience closer to two million events a month.)


Most of those events will be common place. But if you accept the one-in-a-million event definition of a miracle, then according to Littlewood's Law (as it is somewhat grandiosely called) you should experience a "miracle" almost every month because you detect about a million events a month. And once every three or four years, you should experience something as "miraculous" as running into two maternity nurses in a row in a raging snow storm.

Littlewood's proposition doesn't cover all miracles, of course,but it handles most things that people consider miraculous. Thomas Aquinas defined three types of miracles. The highest degree of miracle involved things that can't happen in nature, like the sun standing still or reversing course in the sky. Second degree miracles involve things that don't usually happen in nature, like people presumed dead coming back to life or recovering from various diseases. Third degree miracles are events that can and do happen, but that are sped along faster than you might think, or occur when you don't expect them - a sudden rainstorm ending a drought, or a couple having difficulty conceiving eventually becoming pregnant.

Clearly Littlewood's Law applies to Aquinas' second and third degree miracles. Like Lazarus, even today people seemingly rise from the dead from time to time (usually due to mistaken declarations of death). Many serious diseases are less than a hundred percent fatal, so the fact that someone recovers is a matter of statistics rather than supernatural intervention. And rain coming just when you most need it, but least expect it, happens sometimes simply because weather is complicated and is very, very hard to predict more than a few days in advance.

As helpful as Littlewood is at explaining most miracles, there's not much he can say about the sun reversing it's course. But considering that we now know that it would require the earth's rotation to reverse, leading to catastrophic global destruction, it's pretty clear that things like the Miracle of the Sun must be either optical illusions, or mass hallucinations, or simply mythical - after all, the sun can't do weird things only in one place because, well, Kepler.
http://commons.wikimedia.org/wiki/File:Empirical_Rule.PNG

There is, of course, a mathematical alternative to the word miracle, if you're OK with Littlewood's Law. Considering things that follow normal distributions (which covers lots of everyday circumstances), then events that happen one third of the time are called "one sigma" events (because they are more than one standard deviation, or one sigma, away from the center of the distribution). Two sigma events happen 5% of the time, three sigma events happen 0.3% of the time, and one in a million "miracles" are about 4.9 sigma events.

That fortunate couple caught in the snow storm with a baby on the way?  They experienced way more than a miracle (i.e. a 4.9 sigma event), it was more like a 112% miracle (a 5.5 sigma event)!

I'm sure nothing is going to replace "miracle" in common use when it comes to statistically unlikely events, but maybe we could at least modify the definition in the dictionary.

______________________________________
miracle
[mir-uh-kuh l]

noun

1. A one in a million event.

2. A value more than 4.9 standard deviations away from the mean in normal distribution.


 _________________________________________
 
How's that sound?  Now when you hear someone say, "It's a miracle." You can just reply, "Wow, a one in a million event? That's so lucky for you. I hope you enjoy the dozen or so others you're likely to see this year."
 
 













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