Yes, Virginia, You Can Be 99.99% (Or More!) Certain That 53 Is Prime

Christian apologist William Lane Craig claims the skeptical slogan "extraordinary claims require extraordinary evidence" is contradicted by probability theory, because it actually wouldn't take all that much evidence to convince us that, for example, "the numbers chosen in last night's lottery were 4, 2, 9, 7, 8 and 3." The correct response to this argument is to say that the prior probability of a miracle occurring is orders of magnitude smaller than mere one in a million odds.

This only talks about the probability of the evidence given the truth of the hypothesis, but ignores the probability of the evidence given its falsity. For a variety of reasons, fake claims of miracles are far more common than fake TV announcements of the lottery numbers, which drastically reduces the likelihood ratio you get from the miracle claim relative to the lotto announcement.

The specific miracle also has lower prior probability (miracles are possible+this specific miracle's details), but that's not the only issue.

Generally, you should not be in the habit of doing things that have a 0.1% chance of killing you. Do so on a daily basis, and on average you will be dead in less than three years

Indeed!

It's even worse than that might suggest: 0.999^(3*365.25) = 0.334, so after three years you are almost exactly twice as likely to be dead than alive.

To get 50%, you only need 693 days, or about 1.9 years. Conversely, you need a surprising length of time (6500 days, about 17.8 years) to reduce your survival chances to 0.001.

The field of high-availability computing seems conceptually related. This is often considered in terms of the number of nines - so 'five nines' is 99.999% availability, or <5.3 min downtime a year. It often surprises people that a system can be unavailable for the duration of an entire working day and still hit 99.9% availability over the year. The 'nines' sort-of works conceptually in some situations (e.g. a site that makes money from selling things can't make money for as long as it's unavailable). But it's not so helpful in situations where the cost of an interruption per se is huge, and the length of downtime - if it's over a certain threshold - matters much less than whether it occurs at all. There are all sorts of other problems, on top of the fundamental one that it's very hard to get robust estimates for the chances of failure when you expect it to occur very infrequently. See Feynman's appendix to the report on the Challenger Space Shuttle disaster for amusing/horrifying stuff in this vein.

Very big and very small probabilities are very very hard.

Christian apologist William Lane Craig claims the skeptical slogan "extraordinary claims require extraordinary evidence" is contradicted by probability theory, because it actually wouldn't take all that much evidence to convince us that, for example, "the numbers chosen in last night's lottery were 4, 2, 9, 7, 8 and 3." The correct response to this argument is to say that the prior probability of a miracle occurring is orders of magnitude smaller than mere one in a million odds.

I'm not sure that response works. Flip a fair coin two hundred times, tell me the results, then show me the video and I'll almost certainly believe you. But if the results were H^200, I won't; I'll assume you were wrong or lying about the coin being fair, or something.

H^200 isn't any less likely than any other sequence of two hundred coin flips, but it's still one of the most extraordinary. Extraordinariness just doesn't feel like it's a mere question of prior probability.

Your list actually doesn't go far enough. There is a fourth, and scarier category. Things which would, if possibly render probability useless as a model. "The chance that probabilities don't apply to anything." is in the fourth category. I would also place anything that violates such basic things as the consistency of physics, or the existence of the external world.

For really small probabilities, we have to take into account some sources of error that just aren't meaningful in more normal odds.

For instance, if I shuffle and draw one card from a new deck what is the chance of drawing the ace of spades? I disregard any chance of the deck being defective, any chance of my model of the universe being wrong, and any chance of laws of identity being violated. Any probabilities are eclipsed by the normal probabilities of drawing cards. (category 1)

If I shuffle and draw two cards without replacement from a new deck, what is the chance of them both being aces of spades? Now I have to consider other sources of error. There could have been a factory error or the deck may have been tampered with. (category 2)

If I shuffle and draw one card from a new deck, what is the chance of it being a live tiger? Now I have to consider my model of the universe being drastically wrong. (category 3)

If I shuffle and draw one card from a new deck, what is the chance of it being both the ace of spades and the two of clubs? Not a misprint, and not two cards, but somehow both at the same time. Now I have to consider the law of identity being violated. (category 4)

First, I really like you pointing out the frequent 99% cop out and your partitioning of low-probability events into meaningful categories.

Second, I am not sure that your example with 53 being prime is convincing. It would be more interesting to ask "what unlikely event would break your confidence in 53 being prime?" and estimate the probability of such an event.

This is one of the great reasons to do your math with odds rather than probabilities. (Well, this plus the fact that Bayes' Theorem is especially elegant when formulated in the form of odds ratios.)

There is no reason, save the historical one, that the default mode of thinking is in probabilities (as opposed to odds.) The math works just the same, but for probabilities that are even slightly extreme (even a fair amount less extreme than what is being talked about here), our intuitions about them break down. On the other hand, our intuitions when doing calculations with odds seem to break down a lot less.

When money is on the line, people have more of incentive to avoid errors. That's why, traditionally, gamblers use odds and not probabilities. (e.g. the chance of a horse winning a horse race might be written as "two to nine odds" (or 2:9), and not "eighteen percent chance" (or 0.18). And this example isn't even nearly as extreme as the cases ChrisHallquist talked about, yet still putting it in odds form makes it quite a bit easier to deal with.)

I've never been completely happy with the "I could make 1M similar statements and be wrong once" test. It seems, I dunno, kind of a frequentist way of thinking about the probability that I'm wrong. I can't imagine making a million statements and have no way of knowing what it's like to feel confidence about a statement to an accuracy of one part per million.

Other ways to think of tiny probabilities:

(1) If probability theory tells me there's a 1 in a billion chance of X happening, then P(X) is somewhere between 1 in a billion and P(I calculated wrong), the latter being much higher.

If I were running on hardware that was better at arithmetic, P(I calculated wrong) could be got down way below 1 in a billion. After all, even today's computers do billions of arithmetic operations per second. If they had anything like a one-in-a-billion failure rate per operation, we'd find them much less useful.

(2) Think of statements like P(7 is prime) = 1 as useful simplifications. If I am examining whether 7 is prime, I wouldn't start with a prior of 1. But if I'm testing a hypothesis about something else and it depends on (among other things) whether 7 is prime, I wouldn't assign P(7 is prime) some ridiculously specific just-under-1 probability; I'd call it 1 and simplify the causal network accordingly.

it actually wouldn't take all that much evidence to convince us that, for example, "the numbers chosen in last night's lottery were 4, 2, 9, 7, 8 and 3." The correct response to this argument is to say that the prior probability of a miracle occurring is orders of magnitude smaller than mere one in a million odds.

That doesn't seem right. If somebody tries to convince me that the result of a fair 5 number lottery is 1, 2, 3, 4, 5 I would have a much harder time believing it, but not because the probability is less then one in a million. I think the correct answer is that if the outcome of the lottery wasn't 4, 2, 9, 7, 8, 3 it is very unlikely anybody would try to convince me that the result was exactly that one.

[assume outcome is 4, 2, 9, 7, 8, 3]

Whereas P(outcome) is 1/1 000 000, P(outcome|they tell you the outcome is outcome) is much higher because P(they tell you the outcome is outcome|not outcome) is so much lower then P(they tell you the outcome is outcome|outcome)

In cases like this where we want to drive the probability that something is true as high as possible, you are always left with an incomputable bit.

The bit that can't be computed is - am I sane? The fundamental problem is that there are (we presume) two kinds of people, sane people, and mad people who only think that they are sane. Those mad ones of course come up with mad arguments which show that their sanity is just fine. They may even have supporters who tell them they are perfectly normal - or even hallucinatory ones. How can I show which category I am in? Perhaps instead I am mad, and too mad to know it !

Only mad people can prove that they are sane - the rest of us don't know for sure one way or the other, as every argument in the end returns to the problem that I have to decide whether it's a good argument or not, and whether I am in any position to decide that correctly is the point at issue.

It's quite easy, when trying to prove that 53 must be prime, to get to the position where this problem is the largest remaining issue, but I don't think it's possible to put a number on it. In practice of course I discount the problem entirely as there's nothing I can do about it. I assume I'm fallibly sane rather than barking crazy, and carry on regardless.

Things that have a probability of something like one in a million. Includes many common ways to die that don't involve doing anything most people would regard as especially risky. For example, these stats suggest the odds of a 100 mile car trip killing you are somewhere on the order of one in a million.

I am not entirely sure about this, since I have made a similar mistake in the past, but If I am applying my relatively recent learning of this correctly, I think technically it suggests that if 1 million random people drive 100 miles, one of them will probably die, and based on a quick check of top causes of car accidents, we would also expect that person to be at least one of: Distracted, Speeding, Drunk, Reckless, in the Rain, Drowsy etc...

And if you are currently an undistracted, alert, sober driver who is following all traffic rules in dry weather, your chances of an accident during this particular drive are notably lower (Although I don't know if they are low enough for driving to be safer then a plane because I don't have sufficient statistics on that.)

Edit: Actually I think I'm still insufficiently clear. Technically, we wouldn't be able to expect that they had at least one risk factor without knowing the overall prevalence of those risk factors in the overall population. Admittedly, I did include 'Speeding' as a risk factor and my understanding is that speeding is quite common, (and I did include an etc. to include other non listed risk factors, which would also increase the chance of at least one of them being true) but I haven't actually run the numbers on my expectation above either.

Yeah, that's interesting.

I agree with Eliezer's post, but I think that's a good nitpick. Even if I can't be that certain about 10,000 statements consecutively because I get tired, I think it's plausible that there's 10,000 statements simple arithmetic statements which if I understand, check of my own knowledge, and remember seeing in a list on wikipedia, (which is what I did for 53), that, I've only ever been wrong once on. I find it hard to judge the exact amount, but I definitely remember thinking "I thought that was prime but I didn't really check and I was wrong" but I don't remember thinking "I checked that statement and then it turned out I was still wrong" for something that simple.

Of course, it's hard to be much more certain. I don't know what the chance is that (eg) mathematicians change the definition of prime -- that's pretty unlikely, but similar things have happened before that I thought I was certain of. But rarely.

I agree that you can be 99.99% (or more) certain that 53 is prime but I don't think you can be that confident based only on the arguement you gave.

If a number is composite, it must have a prime factor no greater than its square root. Because 53 is less than 64, sqrt(53) is less than 8. So, to find out if 53 is prime or not, we only need to check if it can be divided by primes less than 8 (i.e. 2, 3, 5, and 7). 53's last digit is odd, so it's not divisible by 2. 53's last digit is neither 0 nor 5, so it's not divisible by 5. The nearest multiples of 3 are 51 (=17x3) and 54, so 53 is not divisible by 3. The nearest multiples of 7 are 49 (=7^2) and 56, so 53 is not divisible by 7. Therefore, 53 is prime.

There are just too many potential errors that could occur in this chain of reasoning. For example, how sure are you that you correctly listed the primes less than 8? Even a mere typo at this stage of the argument could result in an erroneous conclusion.

Anyway just to be clear I do think your high confidence that 53 is prime is justified, but that the argument you gave for it is insufficient in isolation.

Related: Advancing Certainty, and these comments.

I should perhaps include within the text a more direct link to Peter de Blanc's anecdote here:

http://www.spaceandgames.com/?p=27

I won't say "Thus I refute" but it is certainly a cautionary tale.

I think part of what's troubling you about the test is thus: The claim, X has a probability of 10^-30 despite a prior of 50% is roughly equivalent to saying "I have information whose net result is 100 bits of information that X is false" That is certainly a difficult feat, but not really that hard if you put some effort into it (especially when you chose X). The proposed test to verify such a claim, ie making 10^30 similar statements and being wrong only once, would not only be impossible in your lifetime, but would be equivalent to saying "I have 100,000,000,000,000,000,000,000,000,000,000 net bits of information concerning 10^30 questions" Not only will you get defeated by exhaustion and carelessness (including careless choice of X), but your brain just won't hold that much information, which means that such a test would be attempting to predict that in the future, I can acquire that many bits about that many things.

However flawed the test might be, however, the conclusion that you're probably being extremely overconfident is probably still true. I recommend playing one of the calibration games, before you go trusting yourself too much on estimates more than 90%.

I don't think your comment with the lottery is a good example. If there was a lottery last night, then it was going to be some combination of random numbers, with no combination more or less likely then any others. If you come up and tell me "the winning lottery combination last night was X", the odds of you being correct are pretty high; there's really nothing unlikely in that scenario at all. Taking a look at some random number in the real world and thinking about the probability of it is meaningless, since you could be sitting there having the exact same thought no matter what the lottery had ended up being.

If you want to get a better feeling for what a 1 in a million chance feels like, let's say that I come up to you and try to convince you "The numbers that will be chosen in tomorrow night's lottery will be 4, 2, 9, 7, 8 and 3." How easy would it be to convince you of that, before the drawing actually happens?

I feel that the sentence

Suppose you say that you're 99.99% confident that 2 + 2 = 4. Then you have just asserted that you could make 10,000 independent statements, in which you repose equal confidence, and be wrong, on average, around once.

is a little questionable to begin with. What exactly is an "independent" statement in this context? The only way to produce a statement about whether 2 + 2 = 4 holds, is to write a proof that it holds (or doesn't hold). But in a meaningful mathematical system you can't have two independent proofs for the same statement. Two proofs for the same thing are either both right or both wrong, or they aren't proofs in the first place.

[Retracted fully]

I always thought this must be the case from plain observation of thinking; much thinking is "logical", and pure logic is not a suitable model with significant uncertainty. There must be many situations where you're 9.999+ certain in order to make logical thinking useful.

If humans are bad at mental arithmetic, but good at, say, not dying - doesn't that suggest that, as a practical matter, humans should try to rephrase mathematical questions into questions about danger?

E.g. Imagine stepping into a field crisscrossed by dangerous laser beams in a prime-numbers manner to get something valuable. I think someone who had a realistic fear of the laser beams, and a realistic understanding of the benefit of that valuable thing would slow down and/or stop stepping out into suspicious spots.

Quantifying is ONE technique, and it's been used very effectively in recent centuries - but those successes were inside a laboratory / factory / automation structure, not in an individual-rationality context.

This is false modesty. This is assuming the virtue of doubt when none ought exist. Mathematics is one of the few (if not the only) worthwhile thing(s) we have in life that is entirely a priori. We can genuinely achieve 100% certainty. Anything less is to suggest the impossible, or to redefine the world in a way that has no meaning or usefulness.

I could say that I'm not really sure 2+2=4, but it would not make me more intelligent for the doubt, but more foolish. I could say that I'm not sure that 5 is really prime, but it would hinge on redefining '5' or 'prime'. I could posit that if 2+3 reproducibly equaled 4, I would have to change my view of the universe and mathematics, but were I to suggest that argument held any weight, I might as well start believing in God. Define any paradox you like and there will never be a correct answer. The solution is not to accept doubt, but rather to ignore truly unsolvable paradoxes as foolish and useless.

The problem in creating the parallel probability statements is not in the surety, for they would all almost certainly be mathematical as well, but in the daunting task of finding and stating them. This is not reason, this is a threat! "If you assign X probability, are you willing to spend X hours finding parallels?" We react in the negative not due to the reasonability of the rebuttal but rather the daunting task saying yes would hypothetically place upon us. Our chance to perform the task correctly is likely significantly less than that of the probability we have assigned.