Can a rational person believe in miracles?

Rationality and miracles are seemingly in conflict. Rationality is about reason and natural explanations. Miracles are about emotion and supernatural forces. Can the two be reconciled?

My friend sent me a fascinating explanation from The Language of God, a book by Francis Collins, a leader of the Human Genome Project. Collins suggests that rational people can believe in miracles, and he provides an interesting example.

A small introduction to Bayes Theorem

The example uses Bayes Theorem, which is a particularly useful formula in game theory and probability.

Briefly, Bayes Theorem is a formula for computing the probability of an event “conditional” on a base or “prior” hypothesis. In practical applications, Bayes Theorem is useful for determining the likely cause of an event.

An example is that a doctor mentally uses Bayes Theorem when diagnosing a patient. The doctor begins the examination with several “prior” guesses, and updates these probabilities “conditional” on the specific symptoms from follow-up questions. Many times one diagnosis comes out as the likely cause and treatments are administered accordingly. (Here is a more detailed explanation of the math).

Collins suggests we can think the same way after witnessing a truly exceptional event. We can ask whether the event was the result of a natural cause or a divine intervention and use Bayes Theorem to make our guess precise.

A fun example

Here is how Collins makes the idea concrete:

Consider the following example. You have been taken captive by a madman. He gives you a chance to be set free–he will allow you to draw a card from a deck, replace it, shuffle, and draw again. If you draw the ace of spades both times, you will be released.

Skeptical of whether this is even worth attempting, you proceed–and to your amazement you draw the ace of spades twice in a row. Your chains are released and you return home.

Being mathematically inclined, you calculate the chances of this good fortune as 1/52 x 1/52 = 1/2704. A very unlikely event, but it happened. A few weeks later, however, you find out that a benevolent employee of the company that manufactured the playing cards, being aware of the madman’s wager, had arranged that one out of every hundred decks of cards be made up of fifty-two ace of spades.

At this stage, you would consider the likely causes of your drawing two aces of spades. Was it chance from the regular deck, or was it a sure thing from the miracle deck?

So perhaps this was not just a lucky break? Perhaps a knowledgeable and loving being (the employee), unknown to you at the time of your capture, intervened to improve the chances of your release. The likelihood that the deck you drew from was a regular deck of fifty-two different cards was 99/100; the likelihood of a special deck of only aces of spades was 1/100. For those two possible starting points, the “conditional” probabilities of drawing two aces of spades in a row would be 1/2704 and 1, respectively. By Bayes’s Theorem it is now possible to calculate the “posterior” probabilities, and conclude that there is a 96 percent likelihood that the deck of cards you drew from was one of the “miraculous” ones.

In this particular example, a rational person would conclude the good fortune was likely the result of the “miracle.”

Verifying the example

The example can be verified using through a probability tree. Here is how you might compute the odds of the “miracle” deck versus the “regular” deck using Bayes Theorem:

Hedge to example

The example makes it seem like many chance events should be considered miracles.

The misleading part of the example is assigning a very high prior to the miracle deck of 1 in 100. By construction, Collins makes the chance of a miracle deck more than 25 times as likely as drawing two aces of spades. This inflates the chance that drawing two aces came from the miracle deck.

In fact, if we vary the likelihood of the miracle deck, then the probability of that explanation diminishes quickly. If the miracle deck happened with a frequency of one in a million, then the conditional probability falls to less than one percent. In that case, it would be more sensible to conclude the two aces were drawn as a chance outcome from the regular deck.

Though we cannot rule out miracles, we should put their frequency in perspective. Collins elaborates on the example in the book and comes to a similar conclusion. He explains that natural causes do explain a lot so they should be given a high prior probability.

Conclusion

If one accepts the supernatural can exist, then it is logically consistent to be rational and believe in miracles. However, if the supernatural is given a low prior probability, then Bayes Theorem implies that miracles will be rare.

What are your thoughts?