Probability that Miracles Occur: A Bayesian Analysis

One of the most common statements by those professing to know that their interpretation of their specific religious book is true is the assertion that they have observed miracles. They claim that their uncle’s cancer remission is an act of god. Or maybe they claim divine intervention because their bible didn’t burn in a car fire. Or perhaps they see the image of Christ in the stucco wall of their neighbor’s house and proclaim “Miracle!” Whatever the observation they claim it rests on the assumption that they know some divine and supernatural agency was behind it.

Well, no. They have faith. But let’s assume they are 99% certain that the Virgin Mary appearing in toast is a full fledged miracle, leaving behind a mere 1% chance that they are wrong, which is fair—sometimes you gotta hedge your bets at least a little in case the toast turns out to be a fraud.

But that’s part of the point. Maybe it is a fraud. So how should rational people decide whether or not the toast is of divine origin? Bayes’ Theorem!

You might have seen this before on this blog, but if not (and if you aren’t familiar with it) the theorem states that:



In other words, the Bayes’ Theorem calculates the probability of a hypothesis given the observation of new information. In this case I will be testing the probability that miracles occur given the observation of a perceived miracle.

Let H be the hypothesis: Miracles occur.
H’ is the alternate hypothesis: Miracles do not occur.
Let D be the datum: You have observed something that you perceive to be a miracle.
P(H): This is the prior probability that miracles occur (i.e. before you witness a “miracle”). We will set this at 0.001 (or about 1 in 1,000). Before you come at me, let’s be fair; this is being extraordinarily generous.
P(D|H): This is the probability that you observed a “miracle” given that miracles occur. I will be nice and set this at 0.99 (in other words 99% likelihood), which is in keeping with the beliefs in the second paragraph.
P(D|H’): This is the probability that you observed a “miracle” given that miracles do not occur. This is the false positive. I will set this to 0.01 (or 1%) [note that P(D|H) + P(D|H’) must add up to 1].

To sum:
P(H) = 0.001
P(D|H) = 0.99
P(D|H’) = 0.01

Now we can plug this all into the equation above and we come up with the following probability:


In other words if you see something you believe to be a miracle, the Bayes’ Theorem, using my very generous prior probabilities, says that the probability of that event being a miracle is about 9%, with odds 9 to 1 not in your favor.

Now remember that 9% isn’t saying much, considering how nice I was with the prior probability. Considering that in the whole of human history no event has ever occurred that has been verified as a true-to-god miracle under laboratory conditions, the prior probability is actually more like 0.000000001, or about 1 in a billion (and even that is still being generous)! If we plug this into our equation you are going to have to shift that decimal to the left quite a few times.

To sum up, if you see something you believe is a miracle, even with 99% certainty, just check the math. No matter how much you believe it was a true miracle, the probability just isn’t going to do you any favors come time to defend your beliefs. This is not to say you’re wrong—just probably wrong. Very, very probably wrong.

About Rayan Zehn

I'm a political scientist.
This entry was posted in Atheism, Uncategorized and tagged , , , . Bookmark the permalink.

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