Archive for the 'Mathematics' Category

Odds and Ends

Published on March 31, 2008 in Mathematics. 0 Comments Tags: , , .

The place is Vegas. The scene: the lobby of any big hotel. You are walking down a row of slots, past the backs of the acolytes. Hands reach up and down like earthmoving equipment. Large asses envelop the stools they perch on. There is a din of bells and clicking gears, but no one speaks.

Then, up ahead, you hear the gush of coins. A small, mousy woman wearing a red tracksuit and trainers is shoveling a big payout into a plastic bucket. You walk past, then moments later you hear another, smaller delivery. It is the same machine. Registering no emotion, the woman sweeps up her winnings and continues. Entranced, you watch. Nothing happens, for several minutes. The coins are absorbed like lozenges into the belly of a hungry beast. Then her machine once more starts burping coins into its tray. Do you (a) arm-wrestle her out of her place and work that same slot, (b) go tell the management that the machine is fixed, or (c) walk on?

A little later you are standing by the roulette table. Hands with chunky rings push squat piles of chips forward as an offering to the gods of chance. There is a blur of chrome and numbers, and after skittering like a water drop on a griddle, the little ball dives into a slice of the wheel: 17. The next spins ends with a 9. Then a 15, and a 23. With growing amazement, you stare at the wheel. The next spin seems interminable, but after dancing around the edge for nearly thirty seconds, the ball drops. 3! Do you (a) bet the farm that the next number will be even, (b) go tell the management that the croupier is crooked, or (c) walk on?

Finally, you head for the door. A man in shades and a midnight blue suit walks up to you and presses a backpack toward you. Through the half-open zipper you can see wads of bills. By its heft, the backpack must contain a very large sum of money. Do you (a) thank him and leave with the backpack firmly in your grip, (b) engage him in conversation and only accept the gift if his name is not Vinnie or Guido, or do you ignore him and (c) walk on.

The correct answer in each case is (c). Vegas turns many billions of dollars of profit each year thanks to the fact that few people understand probability. Addiction plus ignorance is a potent combination. You can leak it out slowly or void yourself in a series of diarrhetic spasms, but the end result is the same. Unless you can memorize a five-deck shoe or are willing to use radio-controlled devices (and risk the smashed joints that discovery would entail), your fate is sealed. It’s quicker to just go to the cashier as soon as you arrive and hand them a check for all you’ve got.

Independent events have independent odds. What happens next does not depend on what came before. Each roll of the dice, each pull of the lever, each spin of the wheel, is a freshly smelt bouquet of chance.

Here in a nutshell are the rules of probability. Keep them in a safe place, next to your PIN numbers and your spare house key. You never know when you might need them. These rules could (probably) save your life.

Probability ranges from 0 to 1. Zero is something that never happens, and 1 is something that always happens. But things that always happen (night following day) and things that almost never happen (free lunches) are not very interesting. Life occupies the zone between 0 and 1, the realm of the uncertain. You can think of probability as a fraction, as odds, or as a percentage, so ¼, 1 in 4, and 25% are all the same.

Different outcomes of the same event are probabilities that must add up to 1. So if we say that shit happens 3 times out of 10, then, lucky us, shit fails to happen 7 times out of ten. The probability of your next child being a girl is 0.52, and so the probability of it being a boy must be 0.48. Every day when you go into work, there is a 94% chance that nothing will happen, a 5% chance you will get a fat raise, and a 1% chance you will get fired. The sum of these must equal 1, or 100%, unless you subvert nature by lying in bed all day reading trash novels and eating bonbons.

You combine the probability of unrelated events my multiplying them. The odds of rain may be 1 in 2. Perhaps the odds of you breaking your leg on any day are 1 in 500. If you have a teenage daughter, the odds of her coming home with her tongue pierced are 1 in 15. So, on any particular day, the odds of you coming home on a rainy day with a freshly broken leg, having just been fired, to find your daughter with a rod in her mouth, are (1/2) x (1/500) x (1/100) x (1/15) = 1/1,500,000. Less than one in a million. That’s nothing to lose sleep over (which wouldn’t stop me…).

Back to the casino. When the old girl in the tracksuit hits paydirt three times, it’s just the random bunching up of somewhat unlikely events. Imagine rolling a die and getting three consecutive sixes. The odds of it happening are (1/6) x (1/6) x (1/6) or a bit less than 1 in 200. That’s pretty unlikely. But if you roll a die long enough, or pull that lever long enough, eventually it will happen. Meanwhile the casual gambler will have frittered away everything they own.

Also, if the odds of even a modest win are 1 in 200, then in a lobby filled with more than 200 slots, some machine is always paying out. We tend to concentrate on the lucky winner and ignore the sea of losers. So leave the old lady alone. Her machine is not special. If you grabbed it, the chance of it immediately rewarding you with a fistful of dollars are as slim as they are on any slot and at any time.

Same with the roulette wheel — walk on. Five odd numbers in a row sounds like a lot. Actually, the odds are ½ x ½ x ½ x ½ x ½ or 1 in 32. We think that a run of odd numbers makes an even number more likely, but that’s where they have us by the short and curlies. The probability of an even number on that next spin of the wheel is, yessireebob, ½. Try it. Take a coin and flip it repeatedly. You won’t see neatly alternating heads and tails, you will see groups and clusters of heads or tails a lot of the time.

As for the man in the nice suit (who may or may not be a made guy) shoving a bag of money at you; he has nothing to do with probability. Common sense says you should pass. Or ethics, if you want to get all lofty about it.

It’s natural that our critical thinking skills should be compromised in a place like Las Vegas. They are using classic techniques of psychological warfare. Raucous noise, harsh light, cheap drinks and steak dinners at any time of day and night are designed for sensory disorientation. The effect on the human brain is similar to time spent floating in a darkened tank of blood-temperature water. Sensory overload and sensory deprivation — both will leave you yelling for the happy jacket.

Our response to probability is not wholly rational. When people are offered the choice between betting $10 with a 1 in 2 chance of doubling their money or betting $50 with a 1 in 1000 chance of winning $10,000, they will almost always opt for the latter. Yet the second bet is 500 times less likely to pay off. Even if you made both bets over and over, and occasionally won, you would lose money 10 times faster with the second bet.

The lure of the big payoff always overcomes the small prospect of success. A lottery or sweepstakes is an extreme version of this attraction. (Ed McMahon, even as you read this, may be clambering into a minivan with an oversized check made out to YOU!) In a lottery, $1 buys you a shot at $10 million or more. Never mind that you could buy 10 tickets a week for the rest of your life and have no more chance of winning the big prize than of being hit by lightning.

Maybe a lottery is harmless entertainment. A buck is not much money. To inject a bit of logic into the process, you might never buy more than two tickets. Buying that first ticket increases your chances of winning from zero to a miniscule number. Buying the second ticket doubles the chances. But every ticket after that increases the chances by a smaller percentage, with the same amount of money spent.

I, for example, am supposed to be a smarty-pants, logical scientist. Much to my wife’s amusement, every few months I spend some time filling in forms and moving little stickers for on one of the big sweepstakes drawings. Yet I know deep down that I am as likely to win as I am to discover a process for turning plastic into gold. Which does not ease my compulsion to play these tiny odds.

Gambling is a distraction and a sideshow to the main event: the big lottery of life. As hunter-gatherers, we used to roll the dice every day. No calling out for pizza — gathering each meal risked a tussle with a predator. No handy pills or inoculations — each new food was a potential toxin, and each microbe was looking for new hosts. Life is comparably much safer now. Perhaps we need to take risks to feel alive. Las Vegas bets this is the case 365 days a year, and wins. Odds are you won’t be so lucky.

Half of All Penises

Published on February 29, 2008 in Mathematics. 0 Comments

are smaller than average. Not yours, of course. If you are a guy reading this, then yours is no doubt comfortably larger than the mean. Not freak-show huge. Just languid, python-in-the-pants, I’ve-never-had-any-complaints big.

But out there in the world, at the mall, with your lunch crowd, at the water cooler. Half of those guys are concealing weapons that are less than deadly. You would never know it to look at them. They make eye contact, give a firm handshake. But stripped past their skivvies, with their one-eyed johnsons blinking in the light, the awful truth is revealed.

This is not rocket science, but basic math. Take any bunch of numbers, add them up, and divide by the number of numbers: that’s the average, the mean. The numbers spread high and low: that’s the bell curve. To be exactly average is to be special. You are far more likely to either above average or below average. The mean is the line drawn in the sand between the amply and the wimpily endowed.

The numbers are staggering. Of the eighty million adult males in this country, fully forty million have below-average-size penises. Think of the marketing possibilities. Counselors and shrinks could fill their schedules until the next millenium. (On the other hand, a shrink might be the last person you would go to.) Even if no more than a small fraction goes the mechanical route, there’s a vast market for implants and extension devices. Worldwide, we are talking about two billion disadvantaged males. It doesn’t matter that guys have been told, patiently, by women throughout history and by gurus with degrees in both sexual function and dysfunction, that size is not the issue. It’s how you use it. It’s the emotions that go with it. What use is a big penis and a lousy personality. Blahblahblah. In the calculus of the playground or the locker room, the undersized is the second ape, condemned to forever kiss the pink ass of the alpha ape while picking lice from his matted fur.

What can be done? We could redefine average. It already happens in the schools. All our children are comfortingly better than average — the Lake Wobegon effect. Even at university, nearly everyone gets A’s and B’s. Professors are numbed into grade inflation by diminished expectations and laziness, while the universities are too firmly locked to the teat of their tuition money to tell parents that their kid is a lousy student.

All around us, the mediocre has been redefined. Drinks at the movies are Large, Extra Large, and Jumbo, even though the Large would not slake a sparrow’s thirst. Software comes in Standard, Special and Deluxe versions, when in fact it should be labeled No Useful Features, Crashes All The Time, and Only Works On A Computer That You Can’t Afford. Nobody wants to drive or buy a small car, so they are “compact,” or “economy” instead.

This is the worst kind of sophistry. Over two hundred years ago, the brainy (and bearded) German mathematician Karl Friedrich Gauss worked out the theory of random variations. The average is always well defined. There are always equal numbers above and below the average. A bell curve is a bell curve is a bell curve.

Or we could obfuscate. A length in millimeters is twenty-five times larger than the same length in inches. That would bring the average well into the hundreds. Any particular number in the metric system would mean very little to most people. Who hasn’t stared blankly at their weight quoted in kilos or their height quoted in meters?

Maybe it’s more useful to question the data. Stripped down to the issue of scientific measurement, we could ask: How many penises were measured to deduce the average? Would any random set of people really agree to this? At what temperature were the measurements made? Was it inside a cozy lab, or outside on a brisk winter morning when all men are created equal? Was the ruler laid along the top or along the bottom of the gristly shaft? How was curvature accounted for? Was the penis erect or non-erect? If erect, was the erection rightfully or non-rightfully gained? (Surely there are bonus points to becoming large while thinking of your spouse, and demerits for needing a lower form of men’s literature.) Isn’t it better to measure volume? We could use the displacement method of Archimedes, who jumped from his bath and ran down the street naked, having made a pleasing discovery.

And who are these people who make such measurements? (This is a field where self-measurement would be notoriously untrustworthy — what man doesn’t have an inflated opinion of himself inflated?) Do they go home to their wives and kids and read the paper and watch TV just like someone who fixes cars or sells stocks? Do they wash their hands before dinner? We imagine they are professionals, with a professional’s light touch. Like the doctor who gives the annual rectal exam and keeps up a light patter about baseball and the weather. The least embarrassing solution would be to accumulate data from morgues. Unfortunately, you don’t know which stiffs may have shrunk from fright as they faced their Maker. Perhaps the equipment settles into a more compact configuration knowing it will not be called on again.

The variation of body parts is mostly random. Nature rolls the dice as the genes roll on through the generations. It’s not clear what larger earlobes or red hair have to do with survival, but natural selection takes the scattershot approach and tries everything. So size “down there” is just a variable drawn from the same genetic grab bag that causes some people to have eyebrows that grow together.

Prediction is risky. Big people generally have big body parts. But statistics and stereotypes may not apply to individuals. Ladies, you can be misled. There are black men out there with little button mushrooms and Asian men with baby’s arms holding apples in their fists. Pity the poor guy with big hands who gets lots of dates but no call backs. And sometimes big boots mean big feet and that’s all.

Mister Rogers was right. Everyone is special in some way, even if it is an obscure talent like curling your tongue or popping your elbows. You may come up short in one area and hit the long ball in another. This sounds like cold comfort for the smaller-than-average; a tag that says “smallish penis, but a great dancer” or “poorly endowed, but cooks a mean Spanish omelet.”

A furtive obsession with size obscures the fact that most penises are actually quite similar. The bell curve peaks near the average and falls steeply to tails on either side. So the great majority of measurements are close to the average and there are few at very high or low values. Does the difference between a batting average of .270 and .280 matter? Is someone who earns $1000 more than you do better than you? Are you better than someone who earns $1000 less?

Maybe the outlook at one of the ends of the distribution is not as bad (or as good) as you think. Suppose you were puny enough that penetration was an abstract concept. You might be able to parlay it into sympathy, like Howard Stern. At the very least, you would discover the extraordinary capabilities of the human tongue. If you were truly gigantic, certainly there would be a career waiting for you in films whose long suit is thrusting and moaning. But what if you were an actuary or dry goods salesman. Isn’t as obnoxious for a man to be defined by his shaft as it is for a woman to be defined by the flesh lobes that precede her into a room. There are also the practical problems: finding a good tailor, protection during exercise, dealing with the sudden demands on blood flow.

And guys? Women are right. Size doesn’t matter. (Up (or down) to a point.) Technique matters. Listening matters. Making an emotional connection matters. But hey, that’s another story.

Depends on Who You Talk To

Published on January 15, 2008 in Mathematics. 0 Comments Tags: , , , , , , .

What do you really think? About anything. Are your opinions your own, or are you influenced by the popular culture? We all drift in an amniotic fluid of inputs from print media, TV, radio and the Internet. We are surrounded by a relentless white noise of marketing and advertising. How much information is contained in this barrage? How much of it can we trust?

The first human societies had “experts” who everyone went to for advice, information, and wisdom. In different parts of the world, these people were called shamans, priests, village elders, or witchdoctors. Since old people had limited use in a subsistence culture, becoming an information specialist was a good career move. Otherwise, you could be pushed out on an ice floe or left in a cave as jackal bait.

Then the city-state was invented, and information became centralized. Rulers used messengers to deliver their proclamations and town criers to read them out. There was no print media because hardly anybody could read. Casual information was exchanged in the tavern or the marketplace. But moving information around was slow and hazardous. There’s an old nursery rhyme about a horse that loses a nail and so loses a shoe and so loses the rider. As a result, a message is lost and the battle is lost and a war is lost and a king loses his throne. Flaky information flow was the norm over most of human history.

Now we live with instant information. News of every tin-pot dictator, every mid-strength earthquake, every celebrity trip to the laundromat, is right at our fingertips. The only people without Web sites are the homeless. But has this really led to wisdom and democracy? As we drown in media input, we need someone to sort it all out for us. The gatekeepers of the information society are as powerful as ever.

A long time ago, if you wanted to gather information, you did the job right. The Domesday Book was a numbing list of person and everything larger than a jam jar in England 1000 years ago. Our modern census is based on the same idea. The Supreme Court has ruled that every person must be counted in a census, even though the government hasn’t figured out how to keep everyone standing still while they count.

These days, the gathering of information relies on sampling. Instead of asking everyone, you ask every other person or every tenth person. You then assume your sample is typical of the larger population and draw conclusions as if you had really asked everyone.

Scientists use sampling all the time, then they use induction to generalize and draw a broad conclusion. Fragments of certain people’s DNA are being used to figure out how the human genome works, but the knowledge will be quite general because all DNA works the same way. The medical advisory that smoking causes cancer is based on studies of a few thousand smokers. This inference is valid for all 20 million smokers, but it is not deterministic: some smokers will die cancer-free and others will die of cancer that is not caused by smoking. Astronomers know that there are about 50 billion galaxies out there, but not because they have counted them all. With large telescopes, they drill “core samples” and then count galaxies in only a few directions. Then they use the fact that our neighborhood in the universe is not special and calculate the number across the entire sky.

Sampling can be tricky. Let’s say about 10% of all people are left-handed. Out of five randomly selected people in a room, it is unlikely that any would be left-handed. Even with ten people, you might get no lefties. To properly measure the fraction of lefties, you need dozens or maybe a hundred people. A small sample can be unreliable.

The Harris and Gallup polls use only a couple of thousand people to infer the intentions of over 100 million voters. Yet whatever the flaws in the sampling techniques, we can be reassured by the fact that in the end, everyone gets to vote. So our form of government must return people who represent the vibrant, multi-cultural spectrum of American life, right? Or is it a statistical anomaly that most of us are represented in Washington by rich, gray, male, tight-assed lawyers and businessmen?

Sampling also depends on who you talk to. Aliens could drop in a room with five people in it and decide that all humans are right-handed. They might visit Beverly Hills and conclude that everyone has a dye job or a boob job. They might visit Manhattan and deduce that in-your-face pugnaciousness was our defining attribute.

So you should be suitably suspicious when you read that “nine out of ten dentists recommend new Dentagleam with Chromium.” Who are the nine dentists? Name them. Did they have to search the whole country for these 9 stiffs or are they truly representative? How many of them have stock in the Dentagleam parent company? How many of them have kids at Stanford or Harvard on Dentagleam scholarships?

But discussing only the techniques of marketing and advertising is like dwelling on characteristics of the elephant without mentioning the great steaming pile it has left behind.

Does anyone really think that $199.95 is a bargain compared to $200? What use is “3 for the price of 2” when each of them is 50% more than it would have cost individually. Any damn fool can jack the base price up then add a big discount sign. Who do they think they’re kidding? Let’s have some truth in advertising for a change. “It’s overpriced. Deal with it.” Or “If it didn’t wear out so quickly, how could we sell more?” Or “Loads of useless features, but we hope you’re too dumb to notice.”

They have used our sports heroes to sell personal hygiene products and junk food. They have stolen the classic rock songs to sell cross-trainers. They have cloaked their schlock in images of beauty and nobility. We just suck it up, paying top dollar for certain clothes and shoes just to get the little logo. Fashion plate or sandwich board?

The major TV networks — ABC, CBS, NBC, and Fox (aka. Larry, Curley, Moe, and Shemp) — control about 30 billion advertising dollars per year. As niche cable companies proliferate, the “Big 4” control a shrinking wedge of the viewing pie. Think of them as balding men fighting over a comb. Still, $30 billion dollars is not exactly chump-change.

This $30 billion is deployed, plus many billions more in costs for developing new shows, based on data from a very modest sample of Nielsen families. These thousand or so households are randomly selected and their viewing habits are tracked for a week. The selection is random but the sampling is carefully designed to mirror the great viewing masses, more Peoria than Pacific Heights.

Networks worry about the methodology as much as they worry about the small sample. Neilsen families used to fill in viewing diaries, but suppose they fibbed about what they watched? They might litter the log with entries like Masterpiece Theater and Agriculture: A 12-Part History, when they were actually glued to Yet More Inane Home Videos and Wrestlemania XXIII. Is it possible that TV ratings make us seem more highbrow than we are?

So the Neilsen Company now delivers a box which sits on top of the TV and automatically records the viewing choice. But wait, say the advertisers! What if the TV is on but they are not watching? The next step is to add a little video camera to the box, which is preprogrammed with digital images of all family members. Clever software can recognize if young Kevin is slouched in front of the tube, or whether Dad is filling his face with Cheetos and Miller Lite. But what if the family so apathetic that the dog controls the remote?

Marketers and pollsters could gather more and more data on larger samples of people. Or they could acknowledge that most people like the same things and go with that flow. They could use their copious data to look for the person who is truly average in their tastes. Someone who could truly represent us all. Imagine elections boiled down to a single voter. Advertisers could concentrate on this single amalgam of all our preferences — Unibuyer. Let Unibuyer make the choices, the rest of us could then live in a thrillingly ad-free world. Just a thought.