The most recent example I’ve found was in a fantastic TED Talk given by Oxford Mathematician, Peter Donnelly. I’ll embed the video below, if you’d like to watch it yourself. One of the situations that Donnelly presents is this: Suppose we have a test which screens for some disease, and which has a 90% rate of accuracy. If we select someone at random and administer the test, and it gives a positive result, then what is the likelihood that this person carries the disease in question?

Whenever someone presents you with a question like this, you must take the first, intuitive answer that you come up with and ignore it completely. These are trick questions, after all. So I paused the playback on my iPod and spent the next ten minutes trying to work this one out as I walked to my office. Try as I might, however, I couldn’t come up with any answer but the obvious 90%. I felt that possibly there wasn’t enough information to answer the question, but I couldn’t figure out why anything beyond the accuracy rate could be relevant. After all, if a test has a 90% accuracy rate, that suggests it will correct nine times out of ten. Regarding how likely it’s results are to be right, I felt that the stated accuracy rate should answer that question. Otherwise what is the point of an accuracy rate? These are trick questions, however, so I had no illusion about being right.

It turns out that my answer was wrong (of course), but my suspicion was correct. There isn’t enough information to answer the question, because in order to correctly evaluate it we have to also know how how common the disease itself is. Hearing Donnelly explain that, I felt that it made sense but still couldn’t quite understand why. He explained that the likelihood of a positive result being correct has to be weighed against the likelihood of it being false. To illustrate this, he suggested that in a population of one million we might have a 1% infection rate, which amounts to 10,000 people carrying the disease. If you test a random person from that group, and your test has a 90% accuracy rate, then one out of every ten test will provide a bogus result. The implication is that 99,000 uninfected people (10% of 99% of one million) will test positive, even though only 10,000 total people carry the disease. And if my math skills, which are typically quite poor, can work this out, that means the likelihood of an accurate positive result is 89.899%, just slightly less than the indicated 90% accuracy rate.

Huh? 89.899%? Isn’t that basically the same thing? If the close numbers confuse you, as they still confuses me, try working it out with much larger, clearer statistics:

Suppose we have a population of 10 billion people. And suppose we know that, among those people, there are only ten individuals who carry some specific disease. If we have a test for that disease, and it’s 90% accurate, then what are the odds that a single, random individual from the population will actually be infected if his test returns a positive? It can’t be 90%, right? While the odds that any given test is correct may be nine out of ten, that information has to be weighed against the astronomically low probability that any random person actually has the disease, which is 1 out of 10,000,000,000. If you tested every single person, you’d be likely to identify nine of the ten infected people, but you would also get about one billion false positives.

If that doesn’t make the issue clearer for you, suppose that those ten infected people eventually die under quarantine. The disease will no longer be present anywhere in the population, though the test remains unchanged. If we continue to administer the test, we’ll get one positive result out of every ten tests, but there will now be zero possibility of any positive result actually being correct. Clearly the number of people who are infected matters a great deal when evaluating how likely it is that a test result is to be reliable.

In his lecture, Donnelly gives a real-world example of this sort of misunderstanding, wherein a woman was convicted of murdering her children due to this very common misunderstanding of probability. Eventually the conviction was overturned, but it’s a frightening case nonetheless. Check out the video below.

87% of lost dogs who are licensed are returned home.

If your dog gets lost, a license tag on your dog’s collar is the fastest way to reunite you and your dog. Even if your dog is microchipped, a license is immediately visible and doesn’t require a special scanning device to read it. All dog licenses are renewable as of April 1st.

It is your responsibility as a dog owner and good citizen to protect and license your dog.

Now, maybe you don’t mind licensing your dog (if you have one). Maybe you even accept that there are legitimate public health and safety reasons for doing this. But hopefully you will agree with me that this poster is a cut-and-dry case of state propaganda. Let’s look at the manipulation that’s taking place here.

(1) It’s a sad-looking yet adorable puppy. Come on.

(2) It’s misleading. It says “a license tag on your dog’s collar is the fastest way to reunite you and your dog.” This is ridiculous. Any kind of tag on the dog’s collar would have the same effect. All it takes is a phone number on a tiny metal sheet. An address helps too. That costs less than $10. There are automated machines in every PetCo that will stamp them out for you. You do not have to register with the government in order to see these benefits. And frankly, pet safety isn’t the reason why states require pet licensing.

(3) It’s manipulative. “It’s your responsibility as a dog owner and a good citizen to protect and license your dog.” Is it? Is it really? The tacit argument here is that it is our responsibility to comply with all the legal minutia that the state’s come up with, even when they are clearly not moral issues of any kind. If a person’s patriotism and citizenship hinges on whether or not they register their terrier, then those things cannot be worth much to begin with.

Seeing these sorts of posters reminds me that it’s never really clear what the government’s role is supposed to be in our lives. Is it meant to protect us from the threat of foreign powers? Is it meant to provide organization and facilitate municipal services? Or is it meant to maintain bureaucratic pet registration databases? Or to spend tens of thousands of our tax dollars on manipulative ad campaigns that make people feel bad for noncompliance with proto-Orwellian laws?

All men are mortal
Socrates is a man
Therefore, Socrates is mortal

That’s air-tight logic, right there. You can argue with either of the premises (maybe some men aren’t mortal, or maybe Socrates isn’t really a man), but if you accept them then you’re essentially bound to accept the consequent as well. But so what? If you know that all men are mortal, and you know that Socrates is a man, then you already know that Socrates is a mortal. You’ve learned nothing at all. No new information is derived from these statements. They don’t give you any knowledge of men, mortality, or even Socrates, apart from that which you already possessed. In fact, you can use the same logic to show this:

I know that all men are mortal
I know that Socrates is a man
Therefore, I know that Socrates is mortal

So what is the point? If these are just arrangements and rearrangements of symbols, why do we value them so much? Why are there twenty types of syllogism, dozens of fallacies, and various (complicated) systems of formal logic? To answer that, consider another, slightly modified version of our overworked syllogism:

I believe that all men are mortal
I believe that Socrates is a man
Therefore, I believe that Socrates is mortal

Given the simplicity of the statements this is probably true, but it’s not at all guaranteed to be. It’s possible–even quite ordinary–for people to believe each premise of an argument, and yet still deftly deny its consequent. Call it cognitive dissonance, or even just confusion. The nature of the human brain is such that it can hold beliefs which are mutually exclusive, provided (it seems) that it keeps them logically separate. That is, it’s rare for a person to explicitly claim both P and Not P at the same time, but it’s commonplace to hold beliefs which, through a more circuitous path, lead to the same situation.

What this example demonstrates is not that the syllogism is faulty, but that our understanding of “belief” is. We generally consider them to be discrete things that reside in our brains somewhere, waiting to be recalled. But the neurology doesn’t really support this. It turns out that the organization of information and behavior in our brains is quite complex. It’s possible for different parts of our brains to believe and behave in entirely different ways. You can see extreme cases of this in split brain syndrome, but it happens in different parts of our brains, in all sorts of different ways. The positions and statements that we finally come to are just the product of all this, as it manifests in our consciousness. What sort of convoluted neurological gymnastics took place on the road to consciousness is anyone’s guess, however.

The value of formal logic, as I see it, is that it allows us to articulate many different propositions, in a linear manner, and establish a synthesis of further belief from them. In a sense, it’s a means of communications–not just with other people (though it’s certainly that as well), but with ourselves. Considering how much of our everyday thinking is a muddle of emotion and assumption, having a methodology for rigorous thought would seem to be quite helpful. Maybe I should think of formal logic like the scientific method of belief.

A woman has two children, one of which is a girl. What is the probability that the other child is also a girl?

This is a little puzzle that I’ve been stewing over for two days now. I encountered it in the autobiography that I’m reading, Born on a Blue Day, by Daniel Tammet. In his book, Tammet, a mathematical savant, recounts how he’s always had a talent for understanding probability, and that most people find the subject unintuitive. I agreed with his sentiment when I read it. I didn’t count myself among the confused, however, until after I read his example problem, and subsequently answered it wrong.

I thought, as probably many do, that the answer was 1-in-2. That was incorrect. The actual answer is 1-in-3, which I still find rather confounding.

I reasoned that there would be a 50-50 chance. All things being equal, any given child has a 1/2 probability of being either a boy or a girl. Previous events have no effect on the probability of future ones, so I thought the odds would be 1/2 regardless. A sizable segment of the population believes otherwise, though. They think that if your first child is male then your next child is more likely to be female. This is false, of course, so I assumed Tammet’s puzzle would only trick people who made this sort of mistake.

Not so! The answer really is 1-in-3, and here’s why: Imagine that you’ve had two children. There are four possible arrangements of their genders, and each of those four possibilities is equally likely:

1. Boy, Boy
2. Boy, Girl
3. Girl, Boy
4. Girl, Girl

Of the four possibilities, three contain at least one girl. And of those three, only one combination contains two girls. So, to answer the original question: of all the women who have two children, and who have at least one girl, one in three will have two girls.

Q.E.D., right?

That explanation took me a full day to come up with, actually, although I have no doubt that most people would have worked it out much more quickly. In fact, Tammet provides an explanation in his text, although I didn’t want to spoil the puzzle for myself. And it still confused me for another day, because in spite of the demonstrable logic, the answer still didn’t make any sense to me. If I have one daughter, the odds of my next child being a girl are not 1-in-3. They are definitely 1-in-2. As I mentioned above, past events have no effect on future probabilities. It doesn’t matter if I have one daughter, or one son. My next child will always have a 50% chance of being a girl. So how could both of these statements be true?

As it turns out, they’re slightly different statements, although they appear very much alike. The first, the answer to Tammet’s question, gives the probability of there being two daughters, given the knowledge of at least one. The second, my interpretation, gives the probability of having a second daughter when you already have a first. Do you see the difference? My interpretation made the mistake of thinking that this was a question about specific daughters. But it’s not.

It doesn’t ask for the probability of a second child’s gender being female, given the gender of a first. It asks for the probability of a given family’s makeup, given the presence of at least one daughter. The odds of either specific daughter being a girl are still, of course, 1-in-2. Accordingly, the odds of there being a family with two children who are girls are 1-in-4. But given that we can eliminate all families which have all boys (i.e. we know there is at least one girl), those odds rise to 1-in-3. So we never really ask about specific children, and the world of probability makes sense to me once again. Mostly.

For every obscure shepherd kid who becomes a king and every little mustard seed that grows to immeasurable proportions, there are innumerable other small and hidden things that stay hidden and small. There is no reversal of status for them, no eventual success in the ordinary sense, no transformation from an object of derision into an object of admiration, no obvious revelation to be had in their stories. Most of the time, smallness and hiddenness is the whole story, beginning to end.

Small Things Count by J. Mary Luti

Maybe it’s okay to be unremarkable.

That’s not a message you’re likely to encounter often, but it’s worth considering. When I say that it’s okay to be unremarkable, I mean that being unremarkable isn’t the tragedy that we’re lead to believe it is. If we lived our entire lives without being famous, or rich, or powerful; if we never received another promotion, never had our hard work recognized; if we were ignored, or if people thought we were ugly or boring — well, the world would keep spinning all the same. Tens of billions of people have lived and died on this planet, and almost all of them have been basically unremarkable human beings. They weren’t famous or powerful. The didn’t earn a fortune and retire to a tropical paradise. They weren’t beautiful or brilliant. Most of them lived in poverty and ignorance. They suffered through war and hunger and disease. And, of course, every last one of them died.

It’s the same story for those of us who are alive today. Not the being-dead part, of course, but basically everything else. Our daydreams notwithstanding, most of us will never have adoring fans or fancy titles. We’re not going to play before sold-out crowds, or write best-selling novels. For most people, even very talented people, those exclusive honors are simply out of reach. Our unending desire to be better, better, better doesn’t listen to reason, though. It chisels away at our happiness and consumes our lives. Think about your billions of predecessors for a moment. Were they successful? Did they make it? I don’t know. Some did, I suppose, but not in any meaningful way. They’re still dead, and everyone who ever knew them is dead as well.

That’s not a tragedy, though. That’s just life. That’s how it goes. We’re born and we die, and all of the stuff in the middle is eventually washed away by time. We can fill that middle with just about anything, and whatever we end up getting is okay. If we make a tidy income and live comfortably, that’s okay. If we work a nine-to-five job for forty years, and if we retire into mediocrity, that’s alright, too. If we’re overweight and die from heart disease before we turn sixty, that’s fine. Nearly every aspect of our society says the exact opposite, though. It tells us that we should demand more. We should do better, work harder, aim higher. It tells us to never, ever settle. And by-and-large we don’t.

Would we actually be happier, though, if we had more money, more respect, more power? If we lived longer, if we could make ourselves more attractive, smarter? Maybe. Compared to most people, though — just by virtue of living in a developed nation during the 21st century — we already have many of those things. Are we satisfied, then? Most of us aren’t. And our innumerable hordes of unhappy superstars suggest that it wouldn’t matter how much money or fame we acquired; it could never be enough. I think we can be happy with what exactly what we have, though. It doesn’t matter whether we’re high-powered executives or fast-food employees. If we’re able to love ourselves a little, to forgive ourselves for the terrible crime of being an imperfect person, then I think we can find a reasonable amount of happiness in our lives. If we can stop hating ourselves for not living up to our media fantasies, we might be able to recognize the wonderful things that we’ve been missing in our own lives all along.

We have to let go of our pipe dreams to do that. We have to recognize that we’re not going to someday be rich and gorgeous and famous. And it’s not enough to just logically acknowledge this; we have to grasp it at a deeper level. We have to know that not only is it not going to happen, but it’s not important anyway. It wouldn’t provide a lasting happiness, so why are we wasting so much emotional energy on fantasy? When we dream of being someone else, we’re sending ourselves the message that there’s something wrong with who we really are. We’re saying, “The person I am is not good enough.”

If that’s true for us, though, then it’s true for everyone. None of us is good enough. And every one of those tens of billions of people who came before you and I is a failure, too. They weren’t though. They had jobs they didn’t like. They struggled to understand the world and their place in it. They lost people who they loved. Their lives really weren’t that different from our own. They weren’t failures for being unremarkable. And I don’t think that all the basically unremarkable people alive today — people like you and me — are failures either. I think we’re all okay, even if we don’t often realize it.

Should you buy Michelin tires, or Goodyears? Should your next computer be a Dell, or maybe a Compaq? An informed consumer understands that although there are differences between brands, products that exist in the same market, at the roughly same price, will have roughly the same level of quality. The profit differences, and the better deals that exist between brands will be found only at the margins. There may be issues of personal preference to consider: aesthetic appearance, anecdotal experience, etc., but basically there will be no significant differences over a reasonable period of time.

Advertisements tell a different story, though. If you believe the commercials, buying anything other than a Volvo is asking for your family to be killed in a high-speed collision. That’s what commercials are for, after all. They exist, in part, to make us believe that one company’s product is substantially different than the competition. It doesn’t matter that it’s not true, because we don’t have the time or energy to check. When we go to the supermarket, most of us don’t read the ingredient labels on the food we buy. And when we do, we’re unlikely to compare it against another, similar product. Pharmacies across the country sell Bayer right next to less-expensive, chemically-identical, generic aspirin, and most of us are perfectly happy to pay 50% more because of the advertising.

It’s exactly the same with political candidates. If you believe what the political ads, the editorials, and the cable news pundits say, the mainstream candidates could hardly be more different. What are their policy differences, though? They exist, of course, but are they substantial? Do people really understand what they are? My guess is that most of us don’t, and that they have far more in common than they would like most people to believe. Instead of talking about real, debatable differences, we talk about abstractions and generalities. This one is “ready!” That one represents “change!”

When people go to the booths, we make our decisions based on the same criteria that we use to pick between Coke and Pepsi. We don’t even realize it’s happening either. Most of us genuinely believe that our choice is based on sound logic, or even personal preference. But when tested, we can’t even tell the difference.

The entire electoral process is based on this concept of false-choice. Not only are we ignorant of the influence that meaningless advertising and empty arguments play in our decision-making, but we also only have two acceptable choices: Coke. Or Pepsi.

Some people will say “How about RC Cola, then?” completely oblivious to the hold that dark syrup has over their life. Others, less frequently, will suggest that we consider Ginger Ale as a viable, superior alternative to the cola orthodoxy. But it’s still soda pop, isn’t it? And we know all too well that, once the voting is complete, no one really pays any attention to the Ginger Ale drinkers. It’s those slight outsiders who are often blamed for the outcome of the mainstream contest. Ask anyone who voted for Nader. The system encourages homogeny.

If we actually applied our political process to the selection of beverages, we’d find something like this: The vast majority of participants would choose either Coke or Pepsi. Others would be encouraged, for reasons for pragmatism, to abandoned their true opinions and select one of those two. Because once the votes are tallied, it no longer matters. If Coke wins, you drink Coke. It doesn’t matter if you wanted Pepsi. Whatever the bulk of voters think is best is what you’re stuck with. It doesn’t even matter if you don’t want soda at all. Maybe you’re not thirsty. Maybe you just want a glass of water. Too bad. Coke’s the only thing on the menu, and you’re going to be charged for it whether you drink it or not. If you don’t pay your bill, you’ll be put in jail. And if you refuse to go willingly, someone will be authorized to put you there with the help of a gun.

So I’m not voting. I’d rather not take part in a process which is based on lies and exaggerations, and which preys on the basest, most competitive aspects of our personalities.