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# How Likely the Jesus Miracle Stories?

Christianity makes some fanciful claims: Mary was a virgin when she gave birth to Jesus. Jesus turned water into wine. He raised Lazarus from the dead and was resurrected from the dead himself. He is God, one with the creator of the universe.

One response to these claims is simply to dismiss them. Let’s instead see how probability can be applied to questions like this. Christians and non-Christians won’t easily agree because they won’t agree on the various probabilities. Still, an understanding of conditional probability will give us a powerful analytical tool to at least better understand these claims.

Richard Carrier (whose Skepticon video was helpful in my understanding of this material) says that conditional probability “is the mathematical model for all correct reasoning about empirical claims.”

An Example: Medical Test

Start by imagining a test for a disease that is 95% accurate. That is, it is positive 95% of the time for someone who has the disease and negative 95% of the time for someone who doesn’t. Now imagine a common disease—10% of the population has it, so in a thousand people, 100 have it and 900 don’t.

Now give them the test. For the 100 sick people, the test gives 95 positives and 5 (false) negatives. And for the 900 healthy people, it gives 855 (900 × 0.95) negatives and 45 (false) positives.

Suppose the test says that you have the disease. How worrisome is that?

You’re in one of the two groups of people with positive test results. You’re either one of the 95 who actually do have it or one of the 45 who don’t but got a false positive. The chance that you’re sick is the number of sick people who test positive divided by the total number of positives: 95/(95 + 45) = 0.68.

The probability is 68% that you have the disease.

Let’s recap: what’s the probability that you (or any random person) has the disease? 10%. But what’s the probability given that you have a positive test result? It’s 68%. That’s conditional probability—the likelihood of something given (conditional upon) something else, some additional information.

Make the test ten times more accurate and a positive test results means a 96% chance that you have the disease. Instead—and here’s where it gets interesting—make the disease one tenth as common and your likelihood of having the disease given a positive test result is 16%. Make it very rare—one in a million—and that likelihood becomes just 0.005%.*

Visual Approach to the Same Problem

Let’s explore the original problem but visually.

This tree is just a recap of the previous problem: we start with 1000 people, then divide them into two groups based on what we know initially (the probability of a person being sick is 0.1), and finally process this with new information, the test whose probability of a right answer is 0.95.

Applying Probability to the God Question

Let’s move on to the God question (I’m using an example from Richard Carrier’s video).

We start with 1000 universes, places where we imagine God to exist or not. In step 2, our initial assumption about the God claim is to be generous. Knowing nothing about this “God” guy, let’s start by saying that the likelihood of his existence is 50% (P(G) means “probability of God”). In step 2, this gives us two possibilities, with 500 universes in each.

In step 3, we add our new evidence. In the medical example, the new evidence was the result of a test, and here it’s the existence of evil in our world—birth defects, natural disasters that kill thousands, slavery and other immoral institutions, and so on. This evil exists, and yet no god is doing anything about it. What is the likelihood that a benevolent God could exist but still accept the evil in our world?

We have plenty of examples of benevolent beings: the noblest humans. They’re not perfect, but we could assume that a perfectly benevolent being would be at least as benevolent as a good human. Try to imagine a benevolent human (1) who could prevent bad from happening, (2) wouldn’t be harmed for taking this action, but (3) didn’t do anything. That’s pretty inconceivable. Let’s say that the probability of this happening is one in a million. Let’s be conservative and assign the same probability of standing by and doing nothing to a perfectly benevolent god.

That’s the P(e|G) = 10–6 in the diagram above: the probability (P) of the evidence of evil (e) given (|) the existence of God (G) is one in a million (10–6).

The 500 universes on the left side of the tree have to be divided given the probability of such a god existing given the existence of evil. Only one in a million could have a god (~0 means “almost zero”).

It’s easier on the right side of the tree. The likelihood of evil existing in a godless universe is 1.

Conclusion: the existence of evil makes God very improbable.

But … God Could Have an Excuse

In response, the Christian may say that God has an excuse for not acting. Yes, he’s benevolent, but he’s also omniscient, and our finite minds must simply be unable to understand the justification for his inaction.

That doesn’t help, and the tree shows why.

Consider step 3. The conditional probability is now 1. The apologist assumes some unspecified, inconceivable (by our finite brains) reason why God has his own justifiable reasons for allowing evil. But this means we’re looking for something else. We’ve gone from searching for God (G) to searching for “God who has unspecified, inconceivable reasons to allow evil” (G′).

As you can see from step 2, this simply moves the problem around. We had nothing to go on before, so we just assigned a generous 0.5 probability for God (P(G) = 0.5). But now we have a more refined goal that can be evaluated. Now, we’re looking for a very particular God (G′), a very unlikely God, a one-in-a-million God.

Conclusion: making excuses for God makes him less likely. First you must imagine (despite the lack of evidence) supernatural beings, then those with sufficient power to create the universe (deities), then assume that there are benevolent ones that interact with us, then imagine this one-in-a-million deity who has this inconceivable excuse to allow evil.

The mathematics of conditional probability has been applied here to the question, How likely is God given the existence of evil? We could also ask, How likely is the virgin birth given the existence of other virgin birth stories that preceded Jesus that would’ve been known in Palestine? Or, How likely is the resurrection given the existence of stories of other dying-and-rising gods?

This approach will probably never resolve a debate between a Christian and a non-Christian because they won’t be able to agree on probabilities. However, it does give structure to the argument and highlights the unknowns.

Oh, I know He works in mysterious ways,
but if I worked that mysteriously I’d get fired.
— caption for Bob Mankoff cartoon

Appendix: Bayes’ Theorem

We have been using Bayes’ Theorem, though it is more commonly known as an equation. To see that this tree structured approach is an equivalent (and more intuitive) approach to the equation, let’s convert the medical test example above into equation form.

In that example, we first imagine a population of 1000 people and then (step 2) use the likelihood of the disease (10%) to divide that population into sick and well and then (step 3) further divide those populations into those who got positive and negative test results.

Our goal is P(s | p), the probability (P) of being sick (s) given (|) a positive test result (p). Bayes Theorem says that this is computed as follows:

where ~s = the probability of not being sick.

This looks imposing, but you’re already familiar with these values. Look at the numerator first, a measure of how likely s (being sick) is to be the case:

• P(p | s) = the probability of a positive result given that you’re sick = 0.95 (that is, a likelihood of 95%)
• P(s) = the probability of a random person being being sick = 0.1 (the incidence is 10% in the population)

The denominator measures all possible results, your being sick and your being well. It’s the sum of the numerator (the sick likelihood) and its opposite (the not-sick likelihood), which is composed of:

• P(p | ~s) = the probability of a positive test result given that you’re not sick (that is, a false positive), which is 0.05 (our example was simple, with false positives and false negatives both at 5%, but in the general case they would be different)
• P(~s) = the probability of not being sick = 0.9. This one is not a variable since P(~s) = 1 – P(s).

Put these values into the equation: 0.95×0.1/(0.95×0.1 + 0.05×0.9) = 0.67857. This is what we got above with the simpler and more intuitive 95/(95 + 45).

* Here is the math behind those probabilities:
99.5/(99.5 + 4.5) = 0.957
9.5/(9.5 + 49.5) = 0.161
1/(1 + 20,000) = 0.00005

Photo credit: Wikimedia

• Nox

If we are to suppose a miracle to be something so entirely out of the course of what is called nature, that she must go out of that course to accomplish it, and we see an account given of such miracle by the person who said he saw it, it raises a question in the mind very easily decided, which is, is it more probable that nature should go out of her course, or that a man should tell a lie? We have never seen, in our time, nature go out of her course; but we have good reason to believe that millions of lies have been told in the same time; it is therefore, at least millions to one, that the reporter of a miracle tells a lie.

• Bob Seidensticker

Thomas Paine’s Age of Reason–great stuff.

• DrewL

This approach to what is “real” would have successfully inhibited us from ever building our knowledge of dark matter, quantum mechanics, string theory, etc. In other words, you’ve got a deeply deeply flawed approach to reality going on here.

Breaking it down:

X with property A would do B. (A benevolent person would not let a particular preventable act of evil occur)
Therefore, Y with property A would do C. (A benevolent deity would not let ANY preventable acts of evil occur)

You’re presuming you know:
X and Y are for all intents and purposes equivalent.
…and B and C are therefore the logical parallels (A good employee whistleblows on polluting the groundwater, therefore a good deity prevents all birth defects, war, genocide, toothaches, particularly long traffic lights, Ticketmaster processing fees, Nickelback, or anything else that inconveniences Bob.)

Those are not presumptions you know empirically or scientifically. So you’re plugging your own faith beliefs into this approach. Do other people get to plug their faith beliefs in, or just yours?

And most troubling and antithetical to a scientific approach toward knowing reality, you’ve added this premise:

If Y with property A does not do C, but instead D, then Y doesn’t exist.

Talk about lawyer-logic-ing your way to a conclusion. If Einstein would have employed your principle, we wouldn’t have gotten his theory of relativity, he would have come back to tell us the universe simply didn’t exist anymore because it wasn’t complying with the consistent outcomes we had expected from previous theories of physics (in other words, not getting outcome C would have been the end of his investigation, as it is with yours).

This is a terrible mode of inquiry or investigation foranything…why apply it to religious beliefs? Other than personal prejudices, that is.

• Bob Seidensticker

drewl:

you’ve got a deeply deeply flawed approach to reality going on here.

We’re talking about Bayes’ Theorem here, right?

You’re presuming you know:

See my argument as helping you out. You could just handwave with no empirical evidence, or you could find parallels between things we do know and probabilities we’re comfortable with and the God question.

Those are not presumptions you know empirically or scientifically.

They’re guesses. We know of benevolent beings, so we assume that the benevolent properties apply to other benevolent beings. Not a bad assumption, since “having nice properties” is pretty much what it means to be a benevolent being, whether a human, a Klingon, or a god.

then Y doesn’t exist.

What did I say doesn’t exist?

In this confused haze of yours, I was able to pick out that you didn’t like the argument. Fair enough–show me how probability should be applied to the God question.

• DrewL

Not a bad assumption, since “having nice properties” is pretty much what it means to be a benevolent being, whether a human, a Klingon, or a god.

“Having nice properties” is different for neighbors (mind-their-own-business but smile at you), parents of toddlers (incredibly controlling/overbearing/protective without allowing autonomy), parents of young adults (hopefully not controlling/overbearing), marathon trainer (probably not respectful of your autonomy, very authoritative and demanding), bosses, waiters, politicians, corporations, nation-states…since all these are different, which one is going to provide a model for what a benevolent deity looks like? And who gets to to make that call in your mathematical fantasy land, just you?

And you didn’t address the issue of exactly how much “evil/inconvenience” such a deity is responsible for alleviating from your life.

Your equation actually works if you simplify it to: “I don’t care for Mondays, there are Mondays in my world, therefore, it is not very likely a God exists who would remove Mondays from my world.” But then we end up simply ruling out gods no one actually believed in anyway. You have very little to say to any follower of an organized religion other than “Hey we don’t believe in some of the same gods!”

Philosophers of religion have dealt with theism and probability quite extensively, do you want to read them? I find it some of the driest and least persuasive philosophy ever written….assigning probabilities to events and propositional statements will never be a completely objective process, so it’s generally lawyer-logic all the way through. If that’s your thing, I could go dig up some books for you.

• Bob Seidensticker

drewl:

And who gets to to make that call in your mathematical fantasy land, just you?

Good call. Reading my posts closely, I’m sure you’ve correctly inferred my goal of being World Dictator where I make all the decisions.

But no, in this case, I invite input. You know of a better way to wrestle with these probabilities, then toss it out for consideration.

That, or you could just whine about it and criticize. Either way.

And you didn’t address the issue of exactly how much “evil/inconvenience” such a deity is responsible for alleviating from your life.

More than he’s done so far.

Imagine Bambi injured by a fire or fall or a run-in with a wolf, slowly dying in pain over the course of several days. Bambi couldn’t be used to educate or inspire any humans because he’s dying in the wilderness. No benevolent being that I can imagine would allow this. (Or, while we’re at it, birth defects or pedophilia or rape. You get the picture.)

Your equation actually works if you simplify it to: “I don’t care for Mondays, there are Mondays in my world, therefore, it is not very likely a God exists who would remove Mondays from my world.”

We start with benevolent beings that we know about. I haven’t seen this from any human sages, so I don’t see how this is relevant.

You missed one question. You said, “then Y doesn’t exist.” What were you talking about? I don’t remember saying anything that could map onto this structure.

• avalon

@Bob,
You recently warned Christians against using weak analogies. So why present one yourself?
Unlike a disease, God has no well-defined attributes (as Drew pointed out). Christians can always wiggle out of any test for God by simply redefining His attributes.
Perhaps it would have been better to stick to the miracle stories in the bible? But even this won’t work. For example, many scholars noticed a man in the NT miracle stories who had the symptoms of epilepsy. But Jesus said he had a demon. And he expelled the demon to cure him. Was Jesus wrong? Is epilepsy caused by demons? Would a test for epilepsy have been positive? Apologists explain this by saying the demon mimicked epilepsy symptoms!
Bottom line, you can’t test for something that’s not defined.

If the probability of God were like the probability of a disease it’d go something like this:
Bob at his yearly check up:
Doc: You have a disease, Bob.
Bob: But, I feel fine. What are the symptoms?
Doc: They’re different for each individual. It’s a real mystery. Doctor Pope says it’s one thing, Doctor Luther says it’s something else, Doctor Calvin disagrees and says something different.
Bob: Is there a test for this disease?
Doc: It’s strange, every time science disproves one symptom the disease mutates into something else. But you definitely have it.
Bob: But I feel fine!
Doc: I believe you have the disease. You may not believe it, but it’s true. I know deep down in your heart you believe it too. Just have faith and you”ll see it’s true too. Once you believe everything will become clear. You’ll start seeing the symptoms when you have faith in the disease.
Bob: What happens if I don’t believe in this mystery disease?
Doc: Then you’ll never see the symptoms. The disease will lay dormant all your life but after you die you’ll suffer a lot.
Bob: And you know this how?
Doc: I have faith, of course.
(Bob starts looking for a new doctor)

avalon

• Bob Seidensticker

avalon:

Christians can always wiggle out of any test for God by simply redefining His attributes.

Agreed. And that was the point of the final chart–that redefining “God” to mean “someone who has unknowable and inherently not-understandable reasons for allowing evil to continue” doesn’t help the Christian position. It simply moves around the probabilities.

I liked your doctor example. Yes, arguing with an apologist is like that. I tried to be clear that I didn’t think that Bayes Theorem would resolve any problems but simply illustrate or highlight some of the issues.

• Ryan

I’ve always loved conditional probability problems.
The classic example I’ve seen is to assume an incidence rate of 5% (rather than your 10%) which gives you 50/50 odds at the same 5% test failure rates. In fact, as long as your test failure rates are equal (for both types of failure) and your incidence rate also matches the failure rate, it will always come out as 50/50 given the condition that the test came out positive in that case. (I’m ignoring the case of a failure/incidence rate >50%, in which case the condition would need to be the inverse for the probability to hold, only because who would use a test that is wrong >50% of the time?)
Of course, having the same rates of Type I (false positive) and Type II (false negative) errors is highly unusual in itself (Working in security, one of the basics is to learn that virtually every security measure addresses primarily one error type and can even often increase the alternate type), as is having an incidence rate that just magically happens to match either error rate, so the exercise is always somewhat academic, but instructive none the less.

• Selah

Bob ,
In these days a simple child like faith is very rare and every thing is relative and nothing is absolute. Someone said ; ” Doubts are as plentiful as blackberries and all hands and lips are stained with them “. Why do people find ways and think up stuff that ” dooms ” their own salvation ?.
If I were doomed to die , if I had a hint of mercy , I’d certainly find a way to be pardoned.
People who reject God and His son are drowning and are pushing the life preserver away! You and others are preaching your own ruin. You are shutting the door of life in your own face.
Yes , Jesus did many miracles and did many other things and if they should be all recorded one by one in detail, even the world itself could not contain and have room for the books that would be written ( John 21:25 ).All that is needed is child like faith and believe !!

• Bob Seidensticker

Selah:

If I were doomed to die , if I had a hint of mercy , I’d certainly find a way to be pardoned.

What do dying and mercy have to do with each other? If you’re afraid of dying, I think what you want is immortality.

People who reject God and His son are drowning and are pushing the life preserver away!

You’re giving me theology. I think I already know the basic story. What I want is evidence that the Christian story is any more true than the other religions’ stories.

All that is needed is child like faith and believe !!

So why not apply childlike faith to some other religion? Why not become a Sikh or Scientologist?

• ZenDruid

So why not apply childlike faith to some other religion? Why not become a Sikh or Scientologist?

It’s way more fun to create your own mythology.

• Kodie

Find and think up ways to doom our own salvation? Push the life preserver away?

I wonder why you think we are thinking up ways to refuse to believe – it’s not that hard to not believe any of it. If you think believing it can save you from some kind of doom, I don’t think it can. I don’t really even know where these fantasies come from or why you think it’s dangerous not to believe them.

• Selah

Bob ,
You say you want evidence but according to Romans 1 : 18-22 you already have it !. No excuse Bob.
Like one preacher said : ” You know in your knower “. Why would I consider man – made false religions like Sikhism and Scientologism. Mr.Guru Nanak is dead and pushing up daisies ! Oh , I forgot he is going to be reincarnated . I’d rather believe what Hebrews 9:27 says : ” and just as it is appointed for all men ONCE to die and after that certain judgment “. I”ll push all my \$\$ to the center of the table and bet on Jesus because the grave could not hold Him and He was seen by His followers. A real man that Jesus !
Scientology no way ! Man is basically good ? Ask 100 people on the street and 99 will tell you they think of themselves as ” good “.Follow up and ask them if they have ever lied , looked at women with lust in their hearts , stolen anything ,and blasphemed the name of God and all of a sudden they ain’t so good. Maybe in their eyes but not in the eyes of God who says :** No , there is no one that is good .
All have sinned and fallen short of the glory of GOD.
In my opinion , Christianity is the only true ” religion ” ? // actually it’s a personal relationship and that’s hard for some to grasp. Why does following JESUS trump all other religions / movements or what ever you have out there these days is that Christianity is a ” done ” religion in that one man died on the cross and shed His blood and washed away all my sins so that there is nothing else I could add or do to His redemptive act on the Cross at Golgotha. He paid the price for me and in my opinion was and is the greatest substitute / 6th man , to use a rounball term , that ever played the game.
As John 19:30 says : ” When Jesus had received the sour wine , He said, It is finished! ” Faith is the assurance , the title deed of things we hope for being the proof of things we do not see and the conviction of their reality ( faith perceiving as real fact what is not revealed to the senses )

• Bob Seidensticker

Selah:

You say you want evidence but according to Romans 1 : 18-22 you already have it !. No excuse Bob.

“since the creation of the world God’s invisible qualities—his eternal power and divine nature—have been clearly seen”

Weird–how can God’s invisible qualities be clearly seen? Wouldn’t they be … invisible?

Simply saying, “Oh, it’s all around you!” does nothing for me.

Why would I consider man – made false religions like Sikhism and Scientologism.

I dunno–for the same reason that you consider manmade false religion like Christianity? Or is this a trick question?

bet on Jesus because the grave could not hold Him and He was seen by His followers.

We have a story that says that. Don’t you need just a little more evidence than that to believe something so incredible?

Follow up and ask them if they have ever lied , looked at women with lust in their hearts , stolen anything ,and blasphemed the name of God and all of a sudden they ain’t so good.

You didn’t ask them if they were perfect, but if they were good.

God (apparently) made us to look at women with lust. Blasphemy is an invented sin to an invented god.

Faith is the assurance , the title deed of things we hope for being the proof of things we do not see and the conviction of their reality ( faith perceiving as real fact what is not revealed to the senses )

What good is faith? If you need faith to believe something, maybe that something is too tenuous to be worthy of belief.

• Virginia Fitzpatrick

I found your medical Bayesian calculations interesting so being a bit prosaic I tried to follow your example to recalculate the probability that if the disease was one tenth as common that the likelihood of having the disease given a positive test result is 1%. My calculations came up with 2%. Maybe I made some wrong assumptions. I assume the phrase that one tenth as common equaled 1/10 of 10% = 1% i.e. only 1 of out of a population of 1000 would have the disease.
then 1/ 1+ .05*999 = 1/50.95 =.01963 . If you assume that .95 person tests positive then the probability is .0187. At any rate Nate Silvers new book has a wonderful graphic of this rare event problem – Figure 8-6.

As for applying Bayes to belief in a Christian God: I read a news story about a mud slide in England killing all the children in a small town school. Instead of rationalizing God’s intent the town folk took a Bayesian approach and quit going to church. Wish I had the numbers involved so I could plug them into your Bayesian formulas above. However, it amazes me that one observation could overcome a significant prior belief, but then maybe skepticism is higher in church goers than I suspect.

• Bob Seidensticker

Virginia:

My calculations are at the very bottom of the appendix.

• Virginia Fitzpatrick

I reviewed the appendix. The second calculation of the medical screens is:
0.5/(0.5 + 49.5)

I found how the 49.5 in the denominator was derived, but I don’t get (0.5) in the numerator. I calculate the number testing “well” when sick as .5. Instead I would put the number 9.5 in the numerator which is the number of people who text sick when they are sick.

i.e. 9.5 / ( 9.5 – 49.5) = 0.161

• Bob Seidensticker

You’re right. Thanks for the correction.

• http://nw-politics.blogspot.com/?spref=fb Virginia Fitzpatrick

Your welcome. It is nice to know at my age I can still figure.

I was thinking tonight of all the other things people believe in because they want to (besides various Gods) e.g. Obama was born in Kenya, Iraq had nuclear weapons, Everett community college is offering a course in how to use Crystals to enhance your life. My sister gave me a gadget to sanitize my tooth brush. When I went to thank her, she said she had no idea if it really worked as promised. I guess we just accept it on faith.

But which is worse – bad science or blind faith? I saw a lot of dubious data analysis at the Delaware high school science fairs I served as a judge for several years. The clinical trials at the Pharmaceutical firms are biased and their statistical techniques could use improvement and should be updated.

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