Yes, it’s still true. The basic principles of mathematics and probability have not changed. Thus, it is surprising to read an article saying that requiring extraordinary evidence for extraordinary claims doesn’t make sense. At least, it’s surprising until you realize the person making the claim is probably an evangelical Christian with an extraordinary claim to promote. So, how should we respond? I’ll provide a short answer and a long answer.

**The Short Answer**

Bayes’ Theorem, which has been proven to be formally valid, tells us that extraordinary claims (things with very low prior probabilities) do indeed require extraordinary evidence. For example, imagine my friend tells me he won the World Series of Poker. I would first be struck by the improbability of such a claim considering the Main Event has over 6,000 entrants and this particular friend isn’t that good at poker. Even so, it’s not impossible and there is evidence which would overcome that initially low probability. Examples of significant probability-raising evidence would include if he had millions of dollars suddenly and if he had a WSOP bracelet. If you want to see this idea demonstrated mathematically, keep reading.** **

**The Long Answer**

This is Bayes’ Theorem:

P(h|e.b) = P(h|b) x P(e|h.b) / [ P(h|b) x P(e|h.b) ] + [ P(~h|b) x P(e|~h.b) ]

Symbol |
Meaning |

P | Probability |

H | Hypothesis |

E | Evidence |

B | Background Knowledge |

| | Given |

~ | Not (or the negation of) |

. | And |

In English, this says, “The probability that a hypothesis is true, given available evidence and background knowledge is equal to [and then you have the equation].” To understand the equation, let’s dig into the details just a bit.

The first term you’ll notice once you read past the = symbol is P(h|b); this is the prior probability. This term is concerned with the probability that a hypothesis is true given your background knowledge. So, when we say that the virgin birth has an incredibly low prior probability, that means that based on everything we know about the world through science, history, etc. this sort of thing doesn’t generally happen. Quite simply, we understand how babies are made, and this isn’t it. Further, in cases where parthenogenesis does actually occur in other animals, the resulting offspring are always female. So, if true, this would seem to be a one-time thing. I have no idea how many humans have ever existed, but let’s say there have been 100 billion. This would make the prior probability 1/100,000,000,000. Note that this is prior to considering any evidence for the case in question, hence the term prior probability.

The second term above is P(e|h.b), meaning the probability that you would have the available evidence given your hypothesis and background knowledge. In other words, would the available evidence be *expected* under the hypothesis? For example, say someone claims they’ve been to the beach for the past few hours. You notice they are sunburned and in their car is a beach towel. These things fit the claim well and would merit a high number. On the other hand, the lack of sun kissed skin and a movie ticket stub from two hours earlier would *not* be the sort of evidence you would expect.

This covers the numerator. It is the prior probability multiplied by the likelihood of the evidence.

Now, as you keep moving from left to right, you’ll notice that the first term in the denominator is the entire numerator repeated. That’s why the theorem is sometimes shortened to P = A / A + B. So, what is B? It is basically the same as discussed above, only for ~H, rather than H. It is the probability that your hypothesis is false and the likelihood of the available evidence given a false hypothesis.

Let’s plug in some numbers using a very low prior probability and see the earlier claim in action.

- P = 0.01 x .9 / (0.01 x .9) + (0.99 x .75) = 1.2%

Here we see a low probability event where the evidence is almost nearly as well explained by negating hypotheses and the probability is slightly raised, but remains very low. Now, let’s slowly decrease the likelihood that the evidence can be explained by alternative hypotheses and watch what happens to the outcome.

- P = 0.01 x .9 / (0.01 x .9) + (0.99 x .50) = 1.8%
- P = 0.01 x .9 / (0.01 x .9) + (0.99 x .25) = 3.5%
- P = 0.01 x .9 / (0.01 x .9) + (0.99 x .10) = 8.3%
- P = 0.01 x .9 / (0.01 x .9) + (0.99 x .01) = 48%

We don’t see a substantial increase in probability until we get into very low ranges of likelihood for the same evidence to be observed on alternative hypotheses. In the last example, there is only a 1% chance that the evidence can be explained by alternative hypotheses. In cases of low prior probability, the evidence must be such that it basically rules out alternative hypotheses to a very high degree.

In other words, extraordinary claims really do require extraordinary evidence.

## 6 comments

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## Mike

November 29, 2011 at 2:52 pm (UTC 0) Link to this comment

So, the Christian website i linked to ran several more posts on this subject and I wanted to comment and address a misunderstanding they have. One of the articles brought up a lottery winner and said it’s vastly improbable to win the lottery but the ticket is sufficient evidence of it even though a ticket is an “ordinary” piece of paper. This is precisely why we need to formalize our thinking here with Bayes. That author is just playing word games about what is extraordinary. But Bayes tells us what we need. In cases with low priors, we need evidence that uniquely is explained by the hypothesis and virtually rules out competing hypotheses. So, a winning lottery ticket does just that. It is expected (actually 100% of the time) under a lottery winning hypothesis and not expected (actually 0%) under a losing hypothesis. If we plugged in the numbers, it would be 100%. So, ironically the author’s counter example demonstrates my point perfectly!

## Anonymous

December 2, 2011 at 9:25 pm (UTC 0) Link to this comment

This argument doesn’t really help you since the article your trying to argue against gave lots of “extraordinary evidence” for the Truth of Christianity. I also wonder how you would explain the prophecies fuflilled by Jesus from the scriptures.

## Mike

December 4, 2011 at 5:58 pm (UTC 0) Link to this comment

Anonymous,

There are many things I take issue with when it comes to these prophecy claims. One of my biggest problems, though, is what we might call the Judaism problem. Why do most Jews, then and now, reject that Jesus fulfilled messiah prophecies? Here is one reason: most of those passages are not even considered prophecies. Go ahead and read a bunch of them in context. They don’t discuss the messiah and are generally about something happening right then, in the past, or in the immediate future.

## J. Quinton

March 20, 2012 at 8:16 pm (UTC 0) Link to this comment

Thought you might be interested in my attempt at using Bayes’ Theorem on the virgin birth of Jesus: http://deusdiapente.blogspot.com/2012/02/bayes-theorem-and-virgin-birth-of-jesus.html

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## Jim Humphreys

December 19, 2017 at 7:16 pm (UTC 0) Link to this comment

The problem with your argument here is that what counts as an “extraordinary claim” is largely subjective. There is no particular reason that we need to attach any special weight to what the atheist considers to be “extraordinary”; there are billions of theists for whom an atheological position would be extraordinary. But aren’t , for example, theistic miracle claims intrinsically “extraordinary” I hear you ask? Well, it depends on which conditional probability one employs. A miracle is certainly extraordinary given that God does not exist, but not extraordinary given that God exists. So whether one finds something extraordinary will depends on whether or not one holds a priori that God exists. And to establish whether it is plausible that God exists (or does not exist) will require a prior (no pun intended) level of argumentation. So your “extraordinary evidence” maxim really doesn’t take us very far- least if all does it provide support for atheism.

## How to use Bayes' Theorem | Foxhole Atheism

December 2, 2011 at 4:55 pm (UTC 0) Link to this comment

[…] I wrote a Bayesean formulation of Carl Sagan’ s famous maxim, ‘Extraordinary claims require extraordinary evidence.’ However, since the aim of that post was not to teach Bayes’ Theorem, but to reply to a […]

## Absence of Evidence is Evidence of Absence | Foxhole Atheism

December 30, 2011 at 1:43 am (UTC 0) Link to this comment

[…] in Bayes’ Theorem. If you did not read these earlier pieces, you may want to go here and here before reading this […]

## Is naturalism a type of faith? | Foxhole Atheism

April 25, 2012 at 9:09 pm (UTC 0) Link to this comment

[…] If you just read that and thought, “WTF is Bayes’ Theorem?” then you may want to start here, here, and here to see my attempts at […]

## Brief Comments on the Resurrection | Foxhole Atheism

February 20, 2015 at 4:07 am (UTC 0) Link to this comment

[…] It turns out prior probability is quite important to consider. As I described in my post on Extraordinary Claims Really Do Require Extraordinary Evidence, it turns out that the prior probability tells us just how strong the evidence of one competing […]