1 Examples and Principles
Two aristocrats are out horseback riding and one challenges the other to see which can come up with the larger number. The second agrees to the contest, concentrates for a few minutes, and proudly announces, "Three." The proposer of the game is quiet for half an hour, then finally shrugs and concedes defeat.
A summer visitor enters a hardware store in Maine and buys a large number of expensive items. The skeptical, reticent owner doesn't say a word as he adds the bill on the cash register. When he's finished,he points to the total and watches as the man counts out $1,528.47. He then methodically recounts the money once, twice, three times. The visitor finally asks if he's given him the right amount of money, to which the Mainer grudgingly responds, "Just barely."
The mathematician G. H. Hardy was visiting his protégé, the Indian mathematician Ramanujan, in the hospital. To make small talk, he remarked that 1729, the number of the taxi which had brought him, was a rather dull number, to which Ramanujan replied immediately, "No, Hardy! No, Hardy! It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways."
BIG NUMBERS, SMALL PROBABILITIES
People's facility with numbers ranges from the aristocratic to the Ramanujanian, but it's an unfortunate fact that most are on the aristocrats' side of our old Mainer. I'm always amazed and depressed when I encounter students who have no idea what the population of the United States is, or the approximate distance from coast to coast, or roughly what percentage of the world is Chinese. I sometimes ask them as an exercise to estimate how fast human hair grows in miles per hour, or approximately how many people die on earth each day, or how many cigarettes are smoked annually in this country. Despite some initial reluctance (one student maintained that hair just doesn't grow in miles per hour),they often improve their feeling for numbers dramatically.
Without some appreciation of common large numbers, it's impossible to react with the proper skepticism to terrifying reports that more than a million American kids are kidnapped each year, or with the proper sobriety to a warhead carrying a megaton of explosive powerthe equivalent of a million tons (or two billion pounds) of TNT.
And if you don't have some feeling for probabilities, automobile accidents might seem a relatively minor problem of local travel, whereas being killed by terrorists might seem to be a major risk when going overseas. As often observed, however, the 45,000 people killed annually on American roads are approximately equal in number to all American dead in the Vietnam War. On the other hand, the seventeen Americans killed by terrorists in 1985 were among the 28 million of us who traveled abroad that yearthat's one chance in 1.6 million of becoming a victim. Compare that with these annual rates in the United States: one chance in 68,000 of choking to death; one chance in 75,000 of dying in a bicycle crash; one chance in 20,000 of drowning; and one chance in only 5,300 of dying in a car crash.
Confronted with these large numbers and with the correspondingly small probabilities associated with them, the innumerate will inevitably respond with the non sequitur, "Yes, but what if you're that one," and then nod knowingly, as if they've demolished your argument with their penetrating insight. This tendency to personalize is, as we'll see, a characteristic of many people who suffer from innumeracy.Equally typical is a tendency to equate the risk from some obscure and exotic malady with the chances of suffering from heart and circulatory disease, from which about 12,000 Americans die each week.
There's a joke I like that's marginally relevant. An old married couple in their nineties contact a divorce lawyer, who pleads with them to stay together. "Why get divorced now after seventy years of marriage? Why not last it out? Why now?" The little old lady finally pipes up in a creaky voice: "We wanted to wait until the children were dead."
A feeling for what quantities or time spans are appropriate in various contexts is essential to getting the joke. Slipping between millions and billions or between billions and trillions should in this sense be equally funny, but it isn't, because we too often lack an intuitive feeling for these numbers. Many educated people have little grasp for these numbers and are even unaware that a million is 1,000,000; a billion is 1,000,000,000; and a trillion, 1,000,000,000,000.
A recent study by Drs. Kronlund and Phillips of the University of Washington showed that most doctors' assessments of the risks of various operations, procedures, and medications (even in their own specialties) were way off the mark, often by several orders of magnitude. I once had a conversation with a doctor who, within approximately twenty minutes, stated that a certain procedure he was contemplating (a) had a one-chance-in-a-million risk associated with it; (b) was 99 percent safe; and(c) usually went quite well. Given the fact that so many doctors seem to believe that there must be at least eleven people in the waiting room if they're to avoid being idle, I'm not surprised at this new evidence of their innumeracy.
For very big or very small numbers, so-called scientific notation is often clearer and easier to work with than standard notation and I'll therefore sometimes use it. There's nothing very tricky about it: 10N is 1 with N zeroes following it, so 104 is 10,000 and 109 is a billion. 10-N is 1 divided by 10N, so 10-4 is 1 divided by 10,000 or .0001 and 10-2 is one hundredth. 4 106 is 4 1,000,000 or 4,000,000; 5.3 108 is 5.3 100,000,000 or 530,000,000; 2 10-3 is 2 1/1,000 or .002; 3.4 10-7 is 3.4 1/10,000,000 or .00000034.
Why don't news magazines and newspapers make appropriate use of scientific notation in their stories? The notation is not nearly as arcane as many of the topics discussed in these media, and it's considerably more useful than the abortive switch to the metric system about which so many boring articles were written. The expression 7.39842 1010 is more comprehensible and legible than seventy-three billion nine hundred and eighty-four million and two hundred thousand.
Expressed in scientific notation, the answers to the questions posed earlier are: human hair grows at a rate of roughly 10-8 miles per hour; approximately 2.5 105 people die each day on earth; and approximately 5 1011 cigarettes are smoked each year in the United States. Standard notation forthese numbers is: .00000001 miles per hour; about 250,000 people; approximately 500,000,000,000 cigarettes.
BLOOD, MOUNTAINS, AND BURGERS
In a Scientific American column on innumeracy, the computer scientist Douglas Hofstadter cites the case of the Ideal Toy Company, which stated on the package of the original Rubik cube that there were more than three billion possible states the cube could attain. Calculations show that there are more than 4 1019 possible states, 4 with 19 zeroes after it. What the package says isn't wrong; there are more than three billion possible states. The understatement, however, is symptomatic of a pervasive innumeracy which ill suits a technologically based society. It's analogous to a sign at the entrance to the Lincoln Tunnel stating: New York, population more than 6; or McDonald's proudly announcing that they've sold more than 120 hamburgers.
The number 4 1019 is not exactly commonplace, but numbers like ten thousand, one million, and a trillion are. Examples of collections each having a million elements, a billion elements, and so on, should be at hand for quick comparison. For example, knowing that it takes only about eleven and a half days for a million seconds to tick away, whereas almost thirty-two years are required for a billion seconds to pass, gives one a better grasp of the relative magnitudes of these two common numbers. What about trillions? Modern Homo sapiens is probably less than 10 trillion seconds old; and the subsequentcomplete disappearance of the Neanderthal version of early Homo sapiens occurred only a trillion or so seconds ago. Agriculture's been here for approximately 300 billion seconds (ten thousand years), writing for about 150 billion seconds, and rock music has been around for only about one billion seconds.
More common sources of such large numbers are the trillion-dollar federal budget and our burgeoning weapons stockpiles. Given a U.S. population of about 250 million people, every billion dollars in the federal budget translates into $4 for every American. Thus, an annual Defense Department budget of almost a third of a trillion dollars amounts to approximately $5,000 per year for a family of four. What have all these expenditures (ours and theirs) bought over the years? The TNT equivalent of all the nuclear weapons in the world amounts to 25,000 megatons, or 50 trillion pounds, or 10,000 pounds for every man, woman, and child on earth. (One pound in a car, incidentally, demolishes the car and kills everyone in it.) The nuclear weapons on board just one of our Trident submarines contain eight times the firepower expended in all of World War II.
To cite some happier illustrations for smaller numbers, the standard I use for the lowly thousand is a section of Veterans Stadium in Philadelphia which I know contains 1,008 seats and which is easy to picture. The north wall of a garage near my house contains almost exactly ten thousand narrow bricks. For one hundred thousand, I generally think of the number of words in a good-sized novel.
To get a handle on big numbers, it's useful tocome up with one or two collections such as the above corresponding to each power of ten, up to maybe 13 or 14. The more personal you can make these collections, the better. It's also good practice to estimate whatever quantity piques your curiosity: How many pizzas are consumed each year in the United States? How many words have you spoken in your life? How many different people's names appear in The New York Times each year? How many watermelons would fit inside the U.S. Capitol building?
Compute roughly how many acts of sexual intercourse occur each day in the world. Does the number vary much from day to day? Estimate the number of potential human beings, given all the human ova and sperm there have ever been, and you find that the ones who make it to actuality are ipso facto incredibly, improbably fortunate.
These estimations are generally quite easy and often suggestive. For example, what is the volume of all the human blood in the world? The average adult male has about six quarts of blood, adult women slightly less, children considerably less. Thus, if we estimate that on average each of the approximately 5 billion people in the world has about one gallon of blood, we get about 5 billion (5 109) gallons of blood in the world. Since there are about 7.5 gallons per cubic foot, there are approximately 6.7 108 cubic feet of blood. The cube root of 6.7 108 is 870. Thus, all the blood in the world would fit into a cube 870 feet on a side, less than 1/200th of a cubic mile!
Central Park in New York has an area of 840acres, or about 1.3 square miles. If walls were built about it, all the blood in the world would cover the park to a depth of something under 20 feet. The Dead Sea on the Israel-Jordan border has an area of 390 square miles. If all the world's blood were put into the Dead Sea, it would add only three-fourths of an inch to its depth. Even without any particular context, these figures are surprising; there isn't that much blood in the world! Compare this with the volume of all the grass, or of all the leaves, or of all the algae in the world, and man's marginal status among life forms, at least volume-wise, is vividly apparent.
Switching dimensions for a moment, consider the ratio of the speed of the supersonic Concorde, which travels about 2,000 miles per hour, to that of a snail, which moves 25 feet per hour, a pace equivalent to about .005 miles per hour. The Concorde's velocity is 400,000 times that of the snail. An even more impressive ratio is that between the speed with which an average computer adds ten-digit numbers and the rate at which human calculators do so. Computers perform this task more than a million times faster than we do with our snail-like scratchings, and for supercomputers the ratio is over a billion to one.
One last earthly calculation that a scientific consultant from M.I.T. uses to weed out prospective employees during job interviews: How long, he asks, would it take dump trucks to cart away an isolated mountain, say Japan's Mount Fuji, to ground level? Assume trucks come every fifteen minutes, twenty-four hours a day, are instantaneously filled withmountain dirt and rock, and leave without getting in each other's way. The answer's a little surprising and will be given later.
GARGANTUAN NUMBERS AND THE FORBES 400
A concern with scale has been a mainstay of world literature from the Bible to Swift's Lilliputians, from Paul Bunyan to Rabelais' Gargantua. Yet it's always struck me how inconsistent these various authors have been in their use of large numbers.
The infant Gargantua (whence "gargantuan") is said to have needed 17,913 cows to supply him with milk. As a young student he traveled to Paris on a mare that was as large as six elephants, and hung the bells of Notre Dame on the mare's neck as jingles. Returning home, he was attacked by cannon fire from a castle, and combed the cannonballs from his hair with a 900-foot-long rake. For a salad he cut lettuces as large as walnut trees, and devoured half a dozen pilgrims who'd hidden among the trees. Can you determine the internal inconsistencies of this story?
The book of Genesis says of the Flood that " ... all the high hills that were under the whole heaven were covered ..." Taken literally, this seems to indicate that there were 10,000 to 20,000 feet of water on the surface of the earth, equivalent to more than half a billion cubic miles of liquid! Since, according to biblical accounts, it rained for forty days and forty nights, or for only 960 hours, the rain must have fallen at a rate of at least fifteen feet per hour, certainlyenough to sink any aircraft carrier, much less an ark with thousands of animals on board.
Determining internal inconsistencies such as these is one of the minor pleasures of numeracy. The point, however, is not that one should be perpetually analyzing numbers for their consistency and plausibility, but that, when necessary, information can be gleaned from the barest numerical facts, and claims can often be refuted on the basis of these raw numbers alone. If people were more capable of estimation and simple calculation, many obvious inferences would be drawn (or not), and fewer ridiculous notions would be entertained.
Before returning to Rabelais, let's consider two hanging wires of equal cross section. (This latter sentence, I'm sure, has never before appeared in print.) The forces on the wires are proportional to their masses, which are proportional to their lengths. Since the areas of the cross sections of the supporting wires are equal, the stresses in the wire, force divided by cross-sectional area, vary as the lengths of the wires. A wire ten times as long as another will have ten times the stress of the shorter one. Similar arguments show that, of two geometrically similar bridges of the same material, the larger one is necessarily the weaker of the two.
Likewise, a six-foot man cannot be scaled up to thirty feet, Rabelais notwithstanding. Multiplying his height by 5 will increase his weight by a factor of 53, while his ability to support this weightas measured by the cross-sectional area of his boneswill increase by a factor of only 52. Elephants are big but at the cost of quite thick legs, while whalesare relatively immune because they're submerged in water.
Although a reasonable first step in many cases, scaling quantities up or down proportionally is often invalid, as more mundane examples also demonstrate. If the price of bread goes up 6 percent, that's no reason to suspect the price of yachts will go up by 6 percent as well. If a company grows to twenty times its original size, the relative proportions of its departments will not stay the same. If ingesting a thousand grams of some substance causes one rat in one hundred to develop cancer, that's no guarantee that ingesting just one hundred grams will cause one rat in one thousand to develop cancer.
I once wrote to a significant minority of the Forbes 400, a list of the four hundred richest Americans, asking for $25,000 in support for a project I was working on at the time. Since the average wealth of the people I contacted was approximately $400 million (4 108, certainly a gargantuan number of dollars) and I was asking for only 1/16,000th of that wealth, I hoped that linear proportionality would hold, reasoning that if some stranger wrote me asking for support of a worthy project of his and asked me for $25, more than 1/16,000th of my own net worth, I would probably comply with his request. Alas, though I received a number of kind responses, I didn't receive any money.
ARCHIMEDES AND PRACTICALLY INFINITE NUMBERS
There is a fundamental property of numbers named after the Greek mathematician Archimedes which states that any number, no matter how huge, can be exceeded by adding together sufficiently many of any smaller number, no matter how tiny Though obvious in principle, the consequences are sometimes resisted, as they were by the student of mine who maintained that human hair just didn't grow in miles per hour. Unfortunately, the nanoseconds used up in a simple computer operation do add up to lengthy bottlenecks on intractable problems, many of which would require millennia to solve in general. It takes some getting accustomed to the fact that the minuscule times and distances of microphysics as well as the vastness of astronomical phenomena share the dimensions of our human world.
It's clear how the above property of numbers led to Archimedes' famous pronouncement that given a fulcrum, a long enough lever, and a place to stand, he alone could physically lift the earth. An awareness of the additivity of small quantities is lacking in innumerates, who don't seem to believe that their little aerosol cans of hairspray could play any role in the depletion of the ozone layer of the atmosphere, or that their individual automobile contributes anything to the problem of acid rain.
The pyramids, impressive as they are, were built a stone at a time over a period very much shorter than the five thousand to fifteen thousand years required to move the 12,000-foot Mount Fuji by truck.A similar but more classic calculation of this type was made by Archimedes, who estimated the number of grains of sand needed to fill up the earth and heavens. Though he didn't have exponential notation, he invented something comparable, and his calculations were essentially equivalent to the following.
Interpreting "the earth and heavens" to be a sphere about the earth, we observe that the number of grains of sand needed to fill it depends on the radius of the sphere and the coarseness of the sand. Assuming there are fifteen grains per linear inch, there are 15 15 per planar inch and 153 grains per cubic inch. Since there are twelve inches per foot, there are 123 cubic inches per cubic foot and thus 153 123 grains per cubic foot. Similarly, there are 153 123 5,2803 grains per cubic mile. Since the formula for the volume of a sphere is pi the cube of the radius, the number of grains of sand needed to fill a sphere of radius one trillion miles (approximately Archimedes' estimate) is pi 1,000,000,000,0003 153 123 5,2803. This equals approximately 1054 grains of sand.
There is a sense of power connected with such calculations which is hard to explain but which somehow involves a mental encompassing of the world. A more modern version is the calculation of the approximate number of subatomic bits that would fill up the universe. This number plays the role of "practical infinity" for computer problems which are solvable but only theoretically.
The size of the universe is, to be a little generous, a sphere about 40 billion light-years in diameter. Tobe even more generous and also to simplify the rough calculation, assume it's a cube 40 billion light-years on a side. Protons and neutrons are about 10-12 centimeters in diameter. The Archimedean question computer scientist Donald Knuth poses is how many little cubes 10-13 centimeters in diameter ( the diameter of these nucleons) would fit into the universe. An easy calculation shows the number to be less than 10125. Thus, even if a computer the size of the universe had working parts that were smaller than nucleons, it would contain fewer than 10125 such parts, and thus computations on problems which require more parts wouldn't be possible. Perhaps surprisingly, there are many such problems, some of them quite ordinary and of practical importance.
A comparably tiny time unit is the amount of time required for light, which travels at 300,000 kilometers per second, to traverse the length of one of the above tiny cubes, whose edges are 10 -13 centimeters. Taking the universe to be about 15 billion years old, we determine that fewer than 1042 such time units have passed since the beginning of time. Thus, any computer calculation which requires more than 1042 steps (each of which is certainly going to require more time than our unit of time) requires more time than the present history of this universe. Again, there are many such problems.
Taking a human being to be spherical and about a meter in diameter (assume a person is squatting), we end with some biologically revealing comparisons that are somewhat easier to visualize. The size of a human cell is to that of a person as a person's sizeis to that of Rhode Island. Likewise, a virus is to a person as a person is to the earth; an atom is to a person as a person is to the earth's orbit around the sun; and a proton is to a person as a person is to the distance to Alpha Centauri.
THE MULTIPLICATION PRINCIPLE AND MOZART'S WALTZES
Now is probably a good time to reiterate my earlier remark that an occasional difficult passage may be safely ignored by the innumerate reader. The next few sections in particular may contain several such passages. The occasional trivial passage likewise may be quite safely ignored by the numerate reader. (Indeed, the whole book may be safely ignored by all readers, but I'd prefer that, at most, only isolated paragraphs will be.)
The so-called multiplication principle is deceptively simple and very important. It states that if some choice can be made in M different ways and some subsequent choice can be made in N different ways, then there are M N different ways these choices can be made in succession. Thus, if a woman has five blouses and three skirts, then she has 5 3 = 15 choices of outfit, since each of the five blouses (B1, B2, B3, B4, B5) can be worn with any of the three skirts (S1, S2, S3), to yield the following fifteen outfits: B1,S1; B1,S2; B1,S3; B2,S1; B2,S2; B2,S3; B3,S1; B3,S2; B3,S3; B4,S1; B4,S2; B4,S3; B5,S1; B5,S2; B5,S3. From a menu with four appetizers, seven entrees, and three desserts, a diner can design4 7 3 = 84 different dinners, assuming he orders each course.
Likewise, the number of possible outcomes when rolling a pair of dice is 6 6 = 36; any of the six numbers on the first die can be combined with any of the six numbers on the second die. The number of possible outcomes where the second die differs from the first is 6 5 = 30; any of the six numbers of the first die can be combined with any of the remaining five numbers on the second die. The number of possible outcomes when rolling three dice is 6 6 6 = 216. The number of outcomes in which the numbers on the three dice differ is 6 5 4 = 120.
The principle is invaluable in calculating large numbers, such as the total number of telephones reachable without dialing an area code, which comes to roughly 8 106, or 8 million. The first position can be filled by any one of eight digits (0 and 1 aren't generally used in the first position), the second position by any one of the ten digits, and so on, up to the seventh position. (There are actually a few more constraints on the numbers and the positions they can fill, which brings the 8 million figure down somewhat.) Similarly, the number of possible license plates in a state whose plates all have two letters followed by four numbers is 262 104. If repetitions are not allowed, the number of possible plates is 26 25 10 9 8 7.
When the leaders of eight Western countries get together for the important business of a summit meetinghaving their joint picture takenthere are 8 7 6 5 4 3 2 1 = 40,320 different ways in which they can be lined up. Why? Out of these 40,320 ways, in how many would President Reagan and Prime Minister Thatcher be standing next to each other? To answer this, assume that Reagan and Thatcher are placed in a large burlap bag. These seven entities (the six remaining leaders and the bag) can be lined up in 7 6 5 4 3 2 1 = 5,040 ways (invoking the multiplication principle once again). This number must then be multiplied by two since, once Reagan and Thatcher are removed from the bag, we have a choice as to which one of the two adjacently placed leaders should be placed first. There are thus 10,080 ways for the leaders to line up in which Reagan and Thatcher are standing next to each other. Hence, if the leaders were randomly lined up, the probability that these two would be standing next to each other is 10,080/ 40,320 = ¼.
Mozart once wrote a waltz in which he specified eleven different possibilities for fourteen of the sixteen bars of the waltz and two possibilities for one of the other bars. Thus, there are 2 1114 variations on the waltz, only a minuscule fraction of which have ever been heard. In a similar vein, the French poet Raymond Queneau once published a book entitled Cent mille milliards de poèmes, which consisted of a sonnet on each of ten pages. The pages were cut to allow each of the fourteen lines of each sonnet to be turned separately, so that any of the ten first lines could be combined with any of the ten second lines, and so on. Queneau claimed that all the resulting 1014 sonnets made sense, although it's safe to say that the claim will never be verified.
People don't generally appreciate how largesuch seemingly tidy collections can be. A sportswriter once recommended in print that a baseball manager should play every possible combination of his twenty-five-member team for one game to find the nine that play best together. There are various ways to interpret this suggestion, but in all of them the number of games is so large that the players would be long dead before the games were completed.
TRIPLE-SCOOP CONES AND VON NEUMANN'S TRICK
Baskin-Robbins ice-cream parlors advertise thirty-one different flavors of ice cream. The number of possible triple-scoop cones without any repetition of flavors is therefore 31 30 29 = 26,970; any of the thirty-one flavors can be on top, any of the remaining thirty in the middle, and any of the remaining twenty-nine on the bottom. If we're not interested in how the flavors are arranged on the cone but merely in how many three-flavored cones there are, we divide 26,970 by 6, to get 4,495 cones. The reason we divide by 6 is that there are 6 = 3 2 1 different ways to arrange the three flavors in, say, a strawberry-vanilla-chocolate cone: SVC, SCV, VSC, VCS, CVS, and CSV. Since the same can be said for any three-flavored cone, the number of such cones is (31 30 29)/(3 2 1) = 4,495.
A less fattening example is provided by the many state lotteries which require the winner to choose six numbers out of a possible forty. If we're concerned with the order in which these six numbers are chosen, then there are (40 39 38 37 36 35) = 2,763,633,600 ways of choosing them. If, however, we are interested only in the six numbers as a collection (as we are in the case of the lotteries) and not in the order in which they are chosen, then we divide 2,763,633,600 by 720 to determine the number of such collections: 3,838,380. The division is necessary since there are 720 = 6 5 4 3 2 1 ways to arrange the six numbers in any collection.
Another example, and one of considerable importance to card players, is the number of possible five-card poker hands. There are 52 51 50 49 48 possible ways to be dealt five cards if the order of the cards dealt is relevant. Since it's not, we divide the product by (5 4 3 2 1), and find that there are 2,598,960 possible hands. Once that number is known, several useful probabilities can be computed. The chances of being dealt four aces, for example, is 48/2,598,960 ( = about 1 in 50,000), since there are forty-eight possible ways of being dealt a hand with four aces corresponding to the forty-eight cards which could be the fifth card in such a hand.
Note that the form of the number obtained is the same in all three examples: (32 30 29)/ (3 × 2 × 1) different three-flavored ice-cream cones; (40 39 38 37 36 35)/(6 5 4 3 2 1) different ways to choose six numbers out of forty; and (52 51 50 49 48)/(5 4 3 2 1) different poker hands. Numbers obtained in this way are called combinatorial coefficients. They arise when we're interested in the number of ways of choosing R elements out of N elements andwe're not interested in the order of the R elements chosen.
An analogue of the multiplication principle can be used to calculate probabilities. If two events are independent in the sense that the outcome of one event has no influence on the outcome of the other, then the probability that they both occur is computed by multiplying the probabilities of the individual events.
For example, the probability of obtaining two heads in two flips of a coin is ½ ½ = ¼ since of the four equally likely possibilitiestail,tail; tail,head; head,tail; head,headone is a pair of heads. For the same reason, the probability of five straight coin flips resulting in heads is (½)5 = since one of the thirty-two equally likely possibilities is five consecutive heads.
Since the probability that a roulette wheel will stop on red is , and since spins of a roulette wheel are independent, the probability the wheel will stop on red for five consecutive spins is (or .024 2. 4%). Similarly, given that the probability that someone chosen at random was not born in July is , and given that people's birthdays are independent, the probability that none of twelve randomly selected people was born in July is () 12 (or .352 - 35.2%). Independence of events is a very important notion in probability, and when it holds, the multiplication principle considerably simplifies our calculations.
One of the earliest problems in probability was suggested to the French mathematician and philosopherPascal by the gambler Antoine Gombaud, Chevalier de Mère. De Mere wished to know which event was more likely: obtaining at least one 6 in four rolls of a single die, or obtaining at least one 12 in twenty-four rolls of a pair of dice. The multiplication principle for probabilities is sufficient to determine the answer if we remember that the probability that an event doesn't occur is equal to 1 minus the probability that it does (a 20 percent chance of rain implies an 80 percent chance of no rain).
Since is the probability of not rolling a 6 on a single roll of a die, ()4 is the probability of not rolling a 6 in four rolls of the die. Hence, subtracting this number from 1 gives us the probability that this latter event (no 6s) doesn't occur or, in other words, of there being at least one 6 rolled in the four tries: 1 ()4 = .52. Likewise, the probability of rolling at least one 12 in twenty-four rolls of a pair of dice is seen to be 1 ()24 = .49.
A more contemporary instance of the same sort of calculation involves the likelihood of acquiring AIDS hetero-sexually It's estimated that the chance of contracting AIDS in a single unprotected heterosexual episode from a partner known to have the disease is about one in five hundred (the average of the figures from a number of studies). Thus, the probability of not getting it from a single such encounter is 499/500. If these risks are independent, as many assume them to be, then the chances of not falling victim after two such encounters is (499/500)2, and after N such encounters (499/500)N. Since (499/ 500)346 is ½, one runs about a 50 percent chance ofnot contracting AIDS by having unsafe heterosexual intercourse every day for almost a year with someone who has the disease (and thus, equivalently, a 50 percent chance of contracting it).
With a condom the risk of being infected from a single unsafe heterosexual episode with someone known to have the disease falls to one in five thousand, and safe sex every day for ten years with such a person (assuming the victim's survival) would lead to a 50 percent chance of getting the disease yourself. If your partner's disease status is not known, but he or she is not a member of any known risk group, the chance per episode of contracting the infection is one in five million unprotected, one in fifty million with a condom. You're more likely to die in a car crash on the way home from such a tryst.
Two opposing parties often decide an outcome by the flip of a coin. One or both of the parties may suspect the coin is biased. A cute little trick utilizing the multiplication principle was devised by mathematician John von Neumann to allow the contestants to use the biased coin and still get fair results.
The coin is flipped twice. If it comes up heads both times or tails both times, it is flipped twice again. If it comes up heads-tails, this will decide the outcome in favor of the first party, and if it comes up tails-heads, this will decide the outcome in favor of the second party. The probabilities of both these outcomes are the same even if the coin is biased. For example, if the coin lands heads 60 percent of the time and tails 40 percent of the time, a heads-tails sequence has probability .6 .4 = .24 and a tails-heads sequence has probability .4 .6 = .24. Thus,both parties can be confident of the fairness of the outcome despite the possible bias of the coin (unless it is crooked in some different way).
An important bit of background intimately connected to the multiplication principle and combinatorial coefficients is the binomial probability distribution. It arises whenever a procedure or trial may result in "success" or "failure" and one is interested in the probability of obtaining R successes in N trials. If 20 percent of all sodas dispensed by a vending machine overflow their cups, what is the probability that exactly three of the next ten will overflow? at most, three? If a family has five children, what is the probability that they will have exactly three girls? at least, three? If one-tenth of all people have a certain blood type, what is the probability that, of the next hundred people we randomly select, exactly eight will have the blood type in question? at most, eight?
Let me derive the answer to the questions about the vending machine, 20 percent of whose sodas overflow their cups. The probability that the first three sodas overflow and the next seven do not is, by the multiplication principle for probability, (.2)3 (.8)7 . But there are many different ways for exactly three of the ten cups to overflow, each way having probability (.2)3 (.8)7. It may be that only the last three cups overflow, or only the fourth, fifth, and ninth cups, and so on. Thus, since there are altogether (10 9 8)/(3 2 1) = 120 ways for us to pick three out of the ten cups (combinatorial coef ficient), the probability of some collection of exactly three cups overflowing is 120 (.2)3 (.8)7.
The probability of at most three cups overflowing is determined by finding the probability of exactly three cups overflowing, which we've done, and adding to it the probabilities of exactly two, one, and zero cups overflowing, which can be determined in a similar way. Happily, there are tables and good approximations which can be used to shorten these calculations.
JULIUS CAESAR AND YOU
Two final applications of the multiplication principleone slightly depressing, the other somewhat cheering. The first is the probability of not being afflicted with any of a variety of diseases, accidents, or other misfortunes. Not being killed in a car accident may be 99 percent certain, while 98 percent of us may avoid perishing in a household accident. Our chances of escaping lung disease may be 95 percent; dementia, 90 percent; cancer, 80 percent ; and heart disease, 75 percent. These figures are merely for illustration, but accurate estimates may be made for a wide range of dire possibilities. While the chances of avoiding any particular disease or accident may be encouraging, the probability of avoiding them all is not. If we multiply all the above probabilities (assuming these calamities are largely independent), the product grows disturbingly small quite quickly: already our chance of not suffering any of the few misfortunes listed above is less than 50 percent. It's a little anxiety-provoking, how this innocuous multiplication principle can make our mortality more vivid.
Now for better news of a kind of immortal persistence. First, take a deep breath. Assume Shakespeare's account is accurate and Julius Caesar gasped "You too, Brutus" before breathing his last. What are the chances you just inhaled a molecule which Caesar exhaled in his dying breath? The surprising answer is that, with probability better than 99 percent, you did just inhale such a molecule.
For those who don't believe me: I'm assuming that after more than two thousand years the exhaled molecules are uniformly spread about the world and the vast majority are still free in the atmosphere. Given these reasonably valid assumptions, the problem of determining the relevant probability is straightforward. If there are N molecules of air in the world and Caesar exhaled A of them, then the probability that any given molecule you inhale is from Caesar is A/N. The probability that any given molecule you inhale is not from Caesar is thus 1 - A /N. By the multiplication principle, if you inhale three molecules, the probability that none of these three is from Caesar is [1 - A/N]3. Similarly, if you inhale B molecules, the probability that none of them is from Caesar is approximately [1 - A/N]B. Hence, the probability of the complementary event, of your inhaling at least one of his exhaled molecules, is 1 - [1 - A/N]B. A, B (each about 1/30th of a mole, or 2.2 1022), and N (about 1044 molecules) are such that this probability is more than .99. It's intriguing that we're all, at least in this minimal sense, eventually part of one another.
Copyright 1988, 2001 by John Allen Paulos