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Ghosts and the art of bullshit maintenance July 30, 2010

Posted by Ezra Resnick in Belief, Science.
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In Zen and the Art of Motorcycle Maintenance, Robert M. Pirsig presents a view of science that I fear many people will be sympathetic to:

After a while [Chris] says, “Do you believe in ghosts?”

“No,” I say.

“Why not?”

“Because they are un-sci-en-ti-fic.”

The way I say this makes John smile. “They contain no matter,” I continue, “and have no energy and therefore, according to the laws of science, do not exist except in people’s minds.”

The whiskey, the fatigue and the wind in the trees start mixing in my mind. “Of course,” I add, “the laws of science contain no matter and have no energy either and therefore do not exist except in people’s minds. It’s best to be completely scientific about the whole thing and refuse to believe in either ghosts or the laws of science. That way you’re safe. That doesn’t leave you very much to believe in, but that’s scientific too.”

“I don’t know what you’re talking about,” Chris says.

“I’m being kind of facetious.”

Chris gets frustrated when I talk like this, but I don’t think it hurts him.

“One of the kids at YMCA camp says he believes in ghosts.”

“He was just spoofing you.”

“No, he wasn’t. He said that when people haven’t been buried right, their ghosts come back to haunt people. He really believes in that.”

“He was just spoofing you,” I repeat.

“What’s his name?” Sylvia says.

“Tom White Bear.”

John and I exchange looks, suddenly recognizing the same thing.

“Ohhh, Indian!” he says.

I laugh. “I guess I’m going to have to take that back a little,” I say. “I was thinking of European ghosts.”

“What’s the difference?”

John roars with laughter. “He’s got you,” he says.

I think a little and say, “Well, Indians sometimes have a different way of looking at things, which I’m not saying is completely wrong. Science isn’t part of the Indian tradition.”

“Tom White Bear said his mother and dad told him not to believe all that stuff. But he said his grandmother whispered it was true anyway, so he believes it.”

He looks at me pleadingly. He really does want to know things sometimes. Being facetious is not being a very good father. “Sure,” I say, reversing myself, “I believe in ghosts too.”

Now John and Sylvia look at me peculiarly. I see I’m not going to get out of this one easily and brace myself for a long explanation.

“It’s completely natural,” I say, “to think of Europeans who believed in ghosts or Indians who believed in ghosts as ignorant. The scientific point of view has wiped out every other view to a point where they all seem primitive, so that if a person today talks about ghosts or spirits he is considered ignorant or maybe nutty. It’s just all but completely impossible to imagine a world where ghosts can actually exist.”

John nods affirmatively and I continue.

“My own opinion is that the intellect of modern man isn’t that superior. IQs aren’t that much different. Those Indians and medieval men were just as intelligent as we are, but the context in which they thought was completely different. Within that context of thought, ghosts and spirits are quite as real as atoms, particles, photons and quants are to a modern man. In that sense I believe in ghosts. Modern man has his ghosts and spirits too, you know.”

“What?”

“Oh, the laws of physics and of logic — the number system — the principle of algebraic substitution. These are ghosts. We just believe in them so thoroughly they seem real.”

“They seem real to me,” John says.

“I don’t get it,” says Chris.

So I go on. “For example, it seems completely natural to presume that gravitation and the law of gravitation existed before Isaac Newton. It would sound nutty to think that until the seventeenth century there was no gravity.”

“Of course.”

“So when did this law start? Has it always existed?”

John is frowning, wondering what I am getting at.

“What I’m driving at,” I say, “is the notion that before the beginning of the earth, before the sun and the stars were formed, before the primal generation of anything, the law of gravity existed.”

“Sure.”

“Sitting there, having no mass of its own, no energy of its own, not in anyone’s mind because there wasn’t anyone, not in space because there was no space either, not anywhere…this law of gravity still existed?”

Now John seems not so sure.

“If that law of gravity existed,” I say, “I honestly don’t know what a thing has to do to be nonexistent. It seems to me that law of gravity has passed every test of nonexistence there is. You cannot think of a single attribute of nonexistence that that law of gravity didn’t have. Or a single scientific attribute of existence it did have. And yet it is still ‘common sense’ to believe that it existed.”

John says, “I guess I’d have to think about it.”

“Well, I predict that if you think about it long enough you will find yourself going round and round and round and round until you finally reach only one possible, rational, intelligent conclusion. The law of gravity and gravity itself did not exist before Isaac Newton. No other conclusion makes sense.

“And what that means,” I say before he can interrupt, “and what that means is that that law of gravity exists nowhere except in people’s heads! It’s a ghost! We are all of us very arrogant and conceited about running down other people’s ghosts but just as ignorant and barbaric and superstitious about our own.”

“Why does everybody believe in the law of gravity then?”

“Mass hypnosis. In a very orthodox form known as ‘education.'” […]

They are just looking at me so I continue: “Laws of nature are human inventions, like ghosts. Laws of logic, of mathematics are also human inventions, like ghosts. The whole blessed thing is a human invention, including the idea that it isn’t a human invention. The world has no existence whatsoever outside the human imagination. It’s all a ghost, and in antiquity was so recognized as a ghost, the whole blessed world we live in. It’s run by ghosts. We see what we see because these ghosts show it to us, ghosts of Moses and Christ and the Buddha, and Plato, and Descartes, and Rousseau and Jefferson and Lincoln, on and on and on. Isaac Newton is a very good ghost. One of the best. Your common sense is nothing more than the voices of thousands and thousands of these ghosts from the past. Ghosts and more ghosts. Ghosts trying to find their place among the living.”

That may sound profound, but in fact it is profoundly stupid. I’m sorry to be the killjoy rationalist who insists on shaking people out of their fuzzy warm postmodern stupor, but it’s important to, you know, stay in touch with reality. Natural laws (like gravity) are not objects that exist out in space like stars and planets; they are generalizations that describe how our world works. The important thing to remember is that the universe works however it works, whether any people understand it or not. Science is merely our best effort to reach such an understanding – and it turns out that you get better results by methodically testing your hypotheses against reality to see which theory best explains the evidence, rather than just making up stuff that sounds cool or makes you feel good.

Of course, it’s possible to think you understand how the world works, and be mistaken. For instance, we know now that Newton’s laws of gravitation are wrong: they are good approximations for how things work in certain special cases, but they do not apply more generally. However, Newton’s laws still predict the movement of planets a whole lot better than the prophecies of shamans, just as antibiotics are a whole lot better at curing infection than voodoo incantations. As Richard Dawkins puts it:

Show me a cultural relativist at 30,000 feet and I’ll show you a hypocrite. . . If you are flying to an international congress of anthropologists or literary critics, the reason you will probably get there – the reason you don’t plummet into a ploughed field – is that a lot of Western scientifically trained engineers have got their sums right.

Humans can believe any number of things and invent any number of laws; but only some of those beliefs are justified by evidence, and only some of those laws really provide an accurate description of the world we live in.

After a while he says, “Do you believe in ghosts?””No,” I say”Why not?””Because they are un-sci-en-ti-fic.”The way I say this makes John smile. “They contain no matter,” I continue, “and have no energy and therefore, according to the laws of science, do not exist except in people’s minds.”

The whiskey, the fatigue and the wind in the trees start mixing in my mind. “Of course,” I add, “the laws of science contain no matter and have no energy either and therefore do not exist except in people’s minds. It’s best to be completely scientific about the whole thing and refuse to believe in either ghosts or the laws of science. That way you’re safe. That doesn’t leave you very much to believe in, but that’s scientific too.”

Is it rational to give in to irrationality? July 24, 2010

Posted by Ezra Resnick in Game theory, Politics, Reason.
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Reuben and Shimon are placed into a small room with a suitcase containing $100,000 of cash. The owner of the suitcase offers them the following: “I’ll give you all the money in the suitcase, but only on the condition that you negotiate and reach an amicable agreement on its division. That’s the only way I will give you the money.”

Reuben, who is a rational person, appreciates the golden opportunity presented to him and turns to Shimon with the obvious suggestion: “Come, you take half the amount, I’ll take the other half, and each of us will go away with $50,000.” To his surprise, Shimon, with a serious look on his face and a determined voice says: “Listen, I do not know what your intentions are with the money, but I’m not leaving this room with less than $90,000. Take it or leave it. I’m fully prepared to go home with nothing.”

Reuben can not believe his ears. What happened to Shimon? he thinks to himself. Why should he get 90%, and I only 10%? He decides to try to talk to Shimon. “Come, be reasonable,” he pleads. “We’re both in this together, and we both want the money. Come let’s share the amount equally and we’ll both come out ahead.”

But the reasoned explanation of his friend does not seem to register on Shimon. He listens attentively to Reuben’s words, but then declares even more emphatically, “There is nothing to discuss. 90-10 or nothing, that’s my final offer!” Reuben’s face turns red with anger. He wants to smack Shimon across his face, but soon reconsiders. He realizes that Shimon is determined to leave with the majority of the money, and that the only way for him to leave the room with any money is to surrender to Shimon’s blackmail. He straightens his clothes, pulls out a wad of bills from the suitcase in the amount of $10,000, shakes hands with Shimon and leaves the room looking forlorn.

Robert Aumann calls this the blackmailer paradox, or the paradox of the extortionist. It is seemingly a paradox because the rational person gets less than the irrational person, suggesting that the most rational thing you can do is be irrational. Aumann takes this as a model for Israeli-Arab negotiations:

Twenty years ago, a brigadier general came here, maybe a major general, who didn’t identify himself. He wanted to talk to me about the negotiations with Syria. He said to me: ‘You know, Prof. Aumann, the Syrians will not give up a single centimeter of land and the reason is that the land is holy for them. Therefore, they will not concede.’ Then I told him about the extortionist. I said to him: ‘The Syrians have succeeded in convincing you because first they convinced themselves.’

What is our problem? That nothing is holy. We don’t manage to convince ourselves that anything is sacred. Not Jerusalem, not the right of return, not even Tel Aviv. We will be prepared to negotiate, and in the end even to give up Tel Aviv. We are rational as can be, and that’s our problem. . . .

There isn’t anything that we can convince the other side is sacred to us, that we’re willing to ‘be killed for it, rather than transgress.’ If there were something like that, then we wouldn’t be in the situation we are in today.

Before considering Aumann’s proposed solution, let us concede that there is indeed a serious problem here which is too often overlooked: the problem of negotiating with someone who is utterly dogmatic and inflexible. If there is truly no evidence or argument that would ever cause our opponent to change his position, then there is no point in talking to him at all, and we will just have to work around him as best we can. Suppose for a moment that the extortionist Shimon is not a human being, but rather a computer, programmed to accept only those divisions where Reuben gets no more than 10% of the money. Should Reuben attempt to argue with the computer screen, trying to convince it to be reasonable? Clearly not. The rational thing to do in such a case is to take the $10,000 and leave, and there is no paradox. The original scenario feels paradoxical because we expect people not to behave like computers, but rather to be reasonable and open to argument. The scary thing, which many good-intentioned people overlook (at their peril), is that there are human beings who have essentially made themselves into robots following a program: people who have surrendered their mind so completely to dogma that they are incapable of entertaining any alternate positions, even for a moment.

What does Aumann suggest we do when faced with such a dogmatic opponent? Apparently, he thinks we should be more dogmatic ourselves! We must distinguish, of course, between genuine dogmatism and “bluffing:” it may be a clever negotiating tactic to initially feign inflexibility so as to improve your end of the bargain, like hagglers in a market. But it seems that Aumann goes further than this, insisting that we must start by convincing ourselves that our goals are “holy” and “sacred.” Some things may indeed be nonnegotiable and worth dying for, but we need to think very carefully about what those things are, and not mimic the dogmatism of our opponents. Aumann’s proposition seems wrong both as a matter of principle and of practice. Take the issue of Jerusalem, for instance. Is maintaining Israeli sovereignty over all of Jerusalem a cause worth dying for? Is it even a just cause? I see no rational basis for such claims. And even if we did convince ourselves that we’re willing to “be killed for it, rather than transgress,” assuming that the other side is just as dogmatic, how exactly is this going to lead to any kind of solution? Even in the face of irrationality, giving up your own rationality is never the answer – even if it somehow allows you to win a battle, you will have lost the war.

This doesn’t mean we should give in to every demand of an extortionist. When placed in the blackmailer paradox against a computer, it makes sense for Reuben to take the $10,000 because that is better for him than getting nothing; however, if the computer demands that Reuben pay an additional $200,000 out of his own pocket, Reuben would have no reason to comply. If giving the extortionist what he wants is worse for us than the alternative, as it often is, then the rational course of action is to turn him down and do the best we can without him. It’s also important to remember that in real life, we usually find ourselves interacting with the same players over and over, so it may be rational for Reuben to leave the first encounter with nothing in order to make Shimon more likely to cooperate in future encounters. This, of course, assumes that Shimon is rational and is capable of changing his original strategy; against absolute dogmatism, unilateral action may be the only option. And in extreme cases, the only alternative to conversation is force. There is no room for negotiation with a suicide bomber.

Dogmatism is a very serious problem, and we must be willing to use force against it when necessary. More importantly, though, we must fight the foundations of dogmatism by promoting rationality, critical thinking, and the free flow of ideas. Not by becoming dogmatists ourselves.

“The Earth will end only when God declares its time to be over” July 20, 2010

Posted by Ezra Resnick in Belief, Politics, Religion.
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This is Illinois congressman John Shimkus, speaking at a 2009 hearing of the U.S. House Subcommittee on Energy and Environment:

Good news! There’s no need to worry about climate change, since we know that “man will not destroy this Earth” – the Bible says so! I won’t even bother going into how inexcusably stupid and irresponsible Shimkus’s position is, because many religious people probably already agree with me that this idiot should never have been elected to any position of power. However, religious moderates (and many secular people as well) seem to think that the phenomenon he exemplifies is not religion’s fault, and that “sensible” religious people hold no responsibility for the misuse of scripture by some dimwit. I think they are wrong about this, and that the situation we are in is quite a bit more sinister than they are willing to admit.

Moderates seem to think that religious beliefs are somehow different from all our other beliefs about the world; that religious faith is a private matter, which can be kept separate from scientific thinking and rational decision making. A politician’s belief in the divinity of the Bible, for instance, need not affect his public policy. This is simply false. There’s no way around it: if you really believe in a personal God who oversees history, who wants us to behave in a certain way, who will send a messiah to right all wrongs at the end of times, and so on, these beliefs will inevitably influence the decisions you make. Our beliefs about how the world really is are what we use to guide our actions.

Therefore, we must ask ourselves how it is possible for educated people at the top ranks of our society in the 21st century to believe such ridiculous things. After all, Shimkus’s proposition is no different from suggesting that we base our national policy on tea-leaf reading, Nostradamus’s prophecies, or Ouija boards. And yet we don’t see many congressmen supporting those ideas – because they would be ridiculed and marginalized and never would have achieved a position of power in the first place. That is a good thing. The problem is that religious beliefs are given a free pass in our society. While moderates realize there is something wrong with Shimkus’s logic, they still insist on respect for his underlying premise: that the Judeo-Chrisitian God exists, and that the Bible is his “infallible, unchanging, perfect” word. They don’t want people’s core religious beliefs criticized. But you can’t have it both ways.

It really matters what people believe. Insofar as you really believe something, it inevitably constrains your thought processes and affects your decision making. That is why we cannot afford not to care what our neighbors believe, religious matters included. Religious beliefs cannot be kept separate from secular beliefs: there is only one realm of reality, which we try to map in our minds. Either the Bible is divine or it isn’t; either the story it tells is factual or it is not. In all areas, we must demand that people provide good reasons for what they believe. People like Shimkus need to be treated exactly like people who believe Elvis is alive: rather than respecting and legitimizing irrational beliefs accepted on faith, they should be criticized and ridiculed, publicly and incessantly, until superstition withers in the bright light of reality.

If anyone reading this happens to live in Illinois, you know where to start.

I am not a label, I am a free man July 16, 2010

Posted by Ezra Resnick in Language, Politics.
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Living in a complicated world, with a deluge of data constantly bombarding us from all sides, generalizations can often be useful. We instinctively categorize everything around us: threatening/harmless, animal/vegetable/mineral, friend/foe, and so on. This often makes the world easier to deal with, but we can sometimes become slaves to our own man-made labels, our thinking constrained by the assumption that reality must conform to our choice of words. This can be the source of much confusion when things just don’t fit our predetermined categories. Often, this happens when we use discrete categories to label a continuous spectrum. A good example is the labeling of biological species, which continues to cause many misunderstandings about evolution. The naive assumption is that every creature that ever lived can be classified as belonging to some discrete species, but species are labels we bestow in hindsight. The boundaries between species are fluid, as all populations of organisms are in constant transition.

The consequences of labeling can be even greater when it comes to classifying people. First of all, we must obviously overcome superficial stereotyping, where we ascribe traits to people based on arbitrary markers like their nationality, heritage, skin color, etc. It seems to me, however, that the liabilities can be just as serious when labeling people by their political or philosophical ideologies, like “Republican” versus “Democrat” or “capitalist” versus “socialist.” Once we have a label for someone, it becomes easy to dismiss any of his positions out of hand without having to actually consider it carefully or present an explicit counterargument – after all, he’s an X and I’m a Y; end of discussion. (Watch Sam Harris telling the participants at the Atheist Alliance International convention why they should not call themselves “atheists,” or “freethinkers,” or “brights,” or “secular humanists” –  or anything.)

More subtly, people’s labels allow them not to think critically about new issues for themselves: they just toe the party line. (“I’m a Republican, so I must be against the Democrat health care plan.”) Our labels really get in the way of honest, rational discourse and decision making. They distract us from the actual issues at hand.

Some will insist: You must be either a right-winger or a left-winger! If you’re not a socialist or a capitalist, then what are you? I’m sorry to be a troublemaker, but my opinion on any and all matters cannot be summed up in one word. I’m just a person who tries to think rationally about issues on an individual basis, with no automatic allegiance to any ideology. You can put that on my label.

Calling spades spades July 11, 2010

Posted by Ezra Resnick in Game theory, Reason.
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I often play online Spades. Spades is a trick-taking card game for four players (like Bridge, Whist, Hearts, etc.), meaning that every round is composed of thirteen tricks in which each player lays one card. The winner of each trick (determined by simple rules) begins the next trick. The first cool thing about Spades is that it’s a partnership game: players seated opposite each other are on the same team. The objective of the game is to be the first team to reach a predetermined score (e.g., 500 points). Points are given at the end of each round based on the number of tricks won by each team relative to their bid – the number of tricks they declared they would take at the start of the round. The scoring rules are what actually determine the character of the game, and the scoring rules for Spades are pretty simple:

  • A team which won at least the number of tricks they bid receives 10 points for each trick they bid, plus 1 point for each additional trick they won (on top of their bid). However, they are also given 1 penalty point (a bag) for each of the additional tricks. For example, if a team bid 3 but won 5 tricks they receive 32 points plus 2 bags. If a team accumulates 10 bags they are penalized 100 points.
  • A team which won less tricks than they bid is said to be set, and they lose 10 points for each trick they bid. For example, if a team bid 3 but won 2 tricks (or 1, or none), they lose 30 points.

From these rules we can extrapolate some basic strategy. Taking even one trick less than you bid is usually much worse than taking a few extra, so you should only bid an amount you are reasonably sure you can get (when in doubt, underbid). Consequently, the sum of a round’s bids is almost always less than 13 – usually between 10 and 12 (and an occasional sum of 8 or 9 is not unheard of). The asymmetry between overshooting and undershooting also means that it’s often worth taking more tricks than you bid in order to set the other team. This is where things get tricky: the question you are always asking yourself in Spades is, “set or bag?” If you think you have a chance of setting the other team, you may try to take as many tricks as you can, but if you think a set is unlikely then it’s probably better not to take any extra tricks and get your opponents to take all the bags. This razor’s edge is where skill comes into play: you may start a round trying to set your opponents, then switch to bagging them once you realize a set is improbable, or vice versa.

Another thing that makes Spades interesting is the partnership aspect. Your optimal play is obviously dependent on your partner’s cards, and it’s essential that the two of you cooperate on the same plan (e.g., set or bag) – but you don’t know what’s in your partner’s hand (and any discussion of your cards or your preferred strategy is forbidden). Therefore, you must try to infer what your partner has and what strategy he is favoring based on his play, and likewise, you must try to signal your intentions to your partner via your actions within the game. Of course, any such signals can be seen by the opponents as well…

Playing online, you will inevitably encounter some annoying players. Some people can’t resist the urge to berate their partner over an unsuccessful round, even when bad luck is actually to blame. A more interesting example of irrationality, however, is opponents who get upset when they feel you are using a strategy that makes the game “not fun.” David Sirlin, in his “Playing to Win” manifesto, calls such players scrubs. In Spades, scrubs are typically revealed when the sum of bids was low (10 or under), and you have skillfully maneuvered your opponents into taking all the bags. They will then accuse you of being “a bagger” or of playing a “bag game,” i.e., purposely underbidding in order to give the other team more bags. Now, I happen to prefer it when the sum of bids is high, and I usually bid aggressively, but even if I were playing a “bag game” on purpose, this complaint would make no sense. We are all presumably trying to win, so we ought to use whatever strategy works best. If being a “bagger” is a winning strategy (which I don’t think it is), wouldn’t I be a fool not to use it? The thing is, some people only know one strategy, and they get annoyed if their opponent somehow causes it to backfire. But that is the whole point of the game. If the strategy you have chosen is not working well against your opponent’s play, you need to come up with a different strategy. Scrubs will claim that always “playing to win” makes the game less fun, but for me, the fun is in trying to anticipate what the other players will do and figure out how to maximize my chances of success in any given situation. If playing the winning strategy really makes the game no fun for you, find a different game. But don’t be a scrub.

Alternative reality July 10, 2010

Posted by Ezra Resnick in Reason, Science, Superstition.
1 comment so far

It’s amazing how many smart, educated people believe that “alternative” therapies like homeopathy are legitimate and effective. This would seem to indicate a disturbing ignorance of basic scientific principles, and a potentially dangerous reliance on uncritical wishful thinking. Why do I say this? Because the only justification available in support of homeopathy is anecdotal: some people who followed homeopathic advice subsequently reported getting better. In the 21st century, do people really not know that this is not a reliable method for determining the effectiveness of treatments? Such anecdotes are a dime a dozen: people are constantly claiming they were cured by voodoo potions, by the touch of a guru, by praying to the virgin Mary, etc. (Incidentally, recall that bloodletting was widely practiced for 2,000 years even though we now know it was usually harmful to patients!) I’m not saying that everyone who claims to have been helped by “alternative” treatments is lying. However, there are many well-documented phenomena that need to be taken into account before  accepting such claims at face value: the placebo effect, various human biases (like remembering the “hits” and disregarding the “misses”), the probability of conditions going into remission on their own, and more. This is why scientists – who want to cure people no less than homeopaths – perform controlled, double-blind experiments which are analyzed statistically before announcing that a remedy works.

Let me be clear: we don’t have to understand how something works in order to confirm that it works. Even though there is no known scientific basis underlying homeopathic claims, if a properly controlled experiment showed that homeopathic remedies produce better results than a placebo, homeopathy would become a science, and we would be justified in recommending homeopathic remedies even without understanding why they are working. However, homeopathic treatments consistently fail in controlled experiments. The British Medical Association recently voted to stop funding of homeopathic treatments by the National Health Service, and demanded that they be labeled “placebo” when sold in pharmacies.

One might argue that even if homeopathy is just a placebo, some people feel it helps them, so what’s the harm? First of all, pursuing “alternative” therapies can prevent or delay people from getting necessary medical treatment. For example, although malaria kills millions of people each year, and although effective prophylactic malaria drugs are available, ten out of ten alternative health clinics consulted by a BBC reporter in 2006 recommended only homeopathic products for someone traveling to western Africa. (Needless to say, there is no scientific evidence that homeopathic remedies are effective in preventing or treating malaria.) Even if “alternative” treatments are only used in addition to scientific medicine, however, there is still a price to be paid. While there may be some cases where it is ethical for a doctor to prescribe a placebo, there is no excuse for supporting an industry based on ignorance, superstition and deception. It’s fundamentally dishonest to make money by taking advantage of the credulity of desperate people – selling them expensive treatments that have not been shown to be any more effective than a sugar pill.

If you’re still unconvinced, please contact me; I would like to interest you in buying a used car…

Study math, get rich July 6, 2010

Posted by Ezra Resnick in Math, Puzzles.
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You receive a letter informing you that you have been selected as one of 32 lucky people around the world to participate in an experiment, and you may win a million dollars! The rules are simple: by midnight tonight, if exactly one of the 32 participants calls a certain phone number (it doesn’t matter who), you will each receive a million dollars. In any other event (more than one person calls the number, or no one calls), none of you will get anything. Assume the offer is genuine and all participants received the letter on the same day, but you have no way of tracking down the others. What should you do?

The solution is rather straightforward. In order to maximize the chances of exactly one participant making the phone call, each participant should decide to make the call with probability 1/32. For instance, you could roll a die with 32 faces (labeled 1 through 32) and make the phone call only if the number 1 comes up. Another option is to toss five coins and make the call only if they all come up heads (25=32). Problem solved. A more interesting question, however, is this: assuming that all the participants use the optimal strategy, what are the odds that they will actually win the money? And what would be the odds of winning if there were a thousand participants? Or a million?

What I like about problems like this is that the mathematics of the solution are quite simple, and yet the result is counter-intuitive, demonstrating (yet again) how bad human intuition about probability is. In this case, I would have guessed that the odds of winning are pretty small, and that they decrease significantly as the number of participants grows. With a million participants, the chances that all the dice rolls will work out so that exactly one participant makes the call seem pretty slim. Well – let’s think it through. Assume for the moment that there are only two participants, so they each toss a coin and make the phone call only if the result is heads. What are the odds of their winning the money? There are four (equally probable) outcomes: either they both get heads, or they both get tails, or the first participant gets heads and the second gets tails, or the first participant gets tails and the second gets heads. In the first of the four outcomes, they will both make the phone call, and in the second, neither of them will – in both cases, they lose. In either of the latter outcomes, however, exactly one of them makes the call and they win. So the odds of winning in this case are 50%. Not bad, but surely the odds decrease rapidly as we add participants?

Assume there are now three participants. If they all follow the strategy we described, there are 27 (33) possible outcomes, and in 12 of them exactly one participant makes the call, so the odds of winning are about 44%. In the general case, let n denote the number of participants, and assume they each use a die with faces labeled 1 through n. The chances of winning due to participant #1 being the only one to make the phone call are 1/n times ((n-1)/n)n-1 (participant #1 must roll a 1, and all the other n-1 participants must roll any of the n-1 values other than 1). The odds of winning due to participant #2 being the only one to make the phone call are the same as for participant #1, as they are for any other participant, so for the total probability of winning we must multiply the previous expression by n, giving the final answer: ((n-1)/n)n-1.

Plugging n=2 or n=3 into this formula gives the results mentioned above; plugging in n=32 gives a winning probability of about 37% – still pretty good! What about larger values of n? Those who know some calculus will recognize that the above formula converges to 1/e as n grows (where e is Euler’s number, namely 2.718…), so in fact, the odds of winning remain close to 37% no matter how large the number of participants. (Interestingly, the number 1/e pops up in many other, seemingly unrelated, probabilistic questions.)

So remember (especially when visiting a casino): don’t trust your intuitions when it comes to probability!