“Charles Sanders Peirce once observed that in no other branch of mathematics is it so easy for experts to blunder as in probability theory.”
Thus began an article in the October 1959 Scientific American by the celebrated math columnist Martin Gardner. In fact, as John Allen Paulos observed in last January’s issue (“Animal Instincts” [Advances]), humans can sometimes be even worse than pigeons at evaluating probabilities.
Paulos, a mathematician at Temple University in Philadelphia, was describing a notoriously tricky problem known as the Monty Hall paradox. A problem so tricky, in fact, that scores of professional mathematicians and statisticians have stumbled on it over the years. Many have repeatedly failed to grasp it even after they were shown the correct solution.
According to an article by New York Times reporter John Tierney that appeared on July 21, 1991, after a writer called Marilyn vos Savant described the Monty Hall problem—and its uncanny solution—in a magazine the year before, she received something like 10,000 letters, most of them claiming they could prove her wrong. “The most vehement criticism,” Tierney wrote, “has come from mathematicians and scientists, who have alternated between gloating at her (’You are the goat!’) and lamenting the nation’s innumeracy.”
Sure enough, after Paulos mentioned the Monty Hall problem in Scientific American, many readers (though nothing in the order of 10,000) wrote to complain that he had gotten everything wrong, or simply to confess their befuddlement.
“Paulos shows a strange lack of understanding of basic conditional probability,” wrote one reader, “and as a result his article is nonsense.” The reader added that Paulos’s blunder shook his trust in the magazine. “What are your procedures for evaluating submitted papers?” he wrote. This reader was a retired statistics professor.
So we at Scientific American thought it might be worthwhile to try and clarify things a bit. What is this Monty Hall business, and what’s so complicated about it?
The Monty Hall problem was introduced in 1975 by an American statistician as a test study in the theory of probabilities inspired by Monty Hall’s quiz show “Let’s Make a Deal.” (Scholars have observed that the Monty Hall problem was mathematically identical to a problem proposed by French mathematician Joseph Bertrand in 1889—as well as to one, called the three-prisoner game, introduced by Gardner in his 1959 piece; more on that later.) Let’s hear the game’s description from Paulos:
A guest on the show has to choose among three doors, behind one of which is a prize. The guest states his choice, and the host opens one of the two remaining closed doors, always being careful that it is one behind which there is no prize. Should the guest switch to the remaining closed door? Most people choose to stay with their original choice, which is wrong—switching would increase their chance of winning from 1/3 to 2/3. (There is a 1/3 chance that the guest’s original pick was correct, and that does not change.) Even after playing the game many times, which would afford ample opportunity to observe that switching doubles the chances of winning, most people in a recent study switched only 2/3 of the time. Pigeons did better. After a few tries, the birds learn to switch every time.
But wait a minute, you say: after Monty opens the door, there are only two options left. The odds then must be 50-50, or 1/2, for each, so that changing choice of door makes no difference.
To understand what’s going on, we must first make some assumptions, because as it is, the problem’s formulation is ambiguous. So, we shall assume that Monty knows where the car is, and that after the player picks one door he always opens one of the remaining two. Moreover, if the player’s first choice was a door hiding a goat, then Monty always opens the door that hid the other goat; but if the player picked the car, Monty picks randomly between the other two doors, both of which hide a goat.
So imagine you are the player. You take your pick: we’ll call it door 1. One-third of the time, this will be the door with the car, and the remaining 2/3 of the time (66.666… percent) it will be one with a goat. You don’t know what you’ve picked, so you should formulate a strategy that will maximize your overall odds of winning.
Let’s say you picked a goat-hiding door. Monty now opens the other goat-hiding door—call that door 2—and asks you if you want to stick to door 1 or switch to door 3 (which is the one hiding the car). Obviously, in this case by switching you’ll win. But remember, this situation happens 2/3 of the times.
The remaining 1/3 of the time, if you switch, you lose, regardless of which door Monty opens next. But if you adopt the strategy of switching always, no matter what, you’re guaranteed to win 2/3 of the time.
Seems easy enough, doesn’t it? If however you happen to know a little bit of probability theory and you pull out your paper and pencil and start calculating, you might start to doubt this conclusion, as one statistically savvy reader did.
(Warning: this post gets a bit more mathy from here on.)
The reader analyzed the problem using conditional probability, which enables you to answer questions of the type “what are the odds of event A happening given that event B has happened?” The conventional notation for the probability of an event A is P(A), and the notation for “probability of A given B” is P(A | B). The formula to calculate the latter is:
P(A | B) = P(both A and B) / P(B)
The reader wrote:
Let A be the event that the prize is behind door 1 (the initially chosen door), and let B be the event that the prize is not behind door 2 (the door that has been opened). Here, A implies B, so P(both A and B) = P(A) = 1/3, while P(B) = 2/3. Thus P(A | B) = (1/3) / (2/3) = 1/2. Contrary to the claim of Prof. Paulos, nothing is gained by switching from door 1 to door 3. Prof. Paulos is mistaken when he says that P(A | B) = P(A) = 1/3.
What is wrong with this reasoning? It seems utterly plausible and in fact it gave me a headache for about an hour. But it is flawed.
The probability of Monty opening one door or the other changes depending on your initial choice as a player. If you picked a door hiding a goat, Monty has no choice: he is forced to open the door hiding the other goat. If, however, you picked the door hiding the car, Monty has to toss a coin (or some such) before he decides which door to open. But in either case, Monty will open a door that does not hide the prize. Thus, the “event that the prize is not behind door 2 (the door that has been opened)” happens with certainty, meaning P(B) = 1.
Thus, when we apply the formula, we get P(A | B) = (1/3) / (1) = 1/3, not 1/2. The probability that the car is behind door 3 is now 2/3, which means you had better switch.
The Monty Hall paradox is mathematically equivalent to a “wonderfully confusing little problem involving three prisoners and a warden,” the one that Gardner introduced in 1959. Here is Gardner:
Three men—A, B and C—were in separate cells under sentence of death when the governor decided to pardon one of them. He wrote their names on three slips of paper, shook the slips in a hat, drew out one of them and telephoned the warden, requesting that the name of the lucky man be kept secret for several days. Rumor of this reached prisoner A. When the warden made his morning rounds, A tried to persuade the warden to tell him who had been pardoned. The warden refused.
“Then tell me,” said A, “the name of one of the others who will be executed. If B is to be pardoned, give me C’s name. If C is to be pardoned, give me B’s name. And if I’m to be pardoned, flip a coin to decide whether to name B or C.”
“But if you see me flip the coin,” replied the wary warden, “you’ll know that you’re the one pardoned. And if you see that I don’t flip a coin, you’ll know it’s either you or the person I don’t name.”
“Then don’t tell me now,” said A. “Tell me tomorrow morning.” The warden, who knew nothing about probability theory, thought it over that night and decided that if he followed the procedure suggested by A, it would give A no help whatever in estimating his survival chances. So next morning he told A that B was going to be executed.
After the warden left, A smiled to himself at the warden’s stupidity. There were now only two equally probable elements in what mathematicians like to call the “sample space” of the problem. Either C would be pardoned or himself, so by all the laws of conditional probability, his chances of survival had gone up from 1/3 to 1/2.
The warden did not know that A could communicate with C, in an adjacent cell, by tapping in code on a water pipe. This A proceeded to do, explaining to C exactly what he had said to the warden and what the warden had said to him. C was equally overjoyed with the news because he figured, by the same reasoning used by A, that his own survival chances had also risen to 1/2.
Did the two men reason correctly? If not, how should each calculate his chances of being pardoned?
Gardner saved the answer for his next column.
[This post first appeared April 15, 2011 on the Observations blog at ScientificAmerican.com]
On April 8, 1911, at the Leiden Cryogenic Laboratory in the Netherlands, Heike Kamerlingh Onnes and his collaborators immersed a mercury capillary in liquid helium and saw the mercury’s electrical resistance drop to nothing once the temperature reached about 3 kelvins, or 3 degrees above absolute zero (around –270 Celsius).
This “superconductivity” was one of the first quantum phenomena to be discovered, although back then quantum theory did not exist. In subsequent decades theoreticians were able to put quantum physics on a solid foundation and explain superconductivity. Since then, researchers have discovered new families of materials that superconduct at higher and higher temperatures: the current record-holder works at a balmy 138 K.
So where’s my maglev train?
Indeed, the promise of superconductors—power grids that waste no energy, computers that run at untold gigahertz of speed without overheating and, yes, trains that levitate over magnetic fields—has not fully materialized.
Still, superconductors have made it possible to build the strong magnets that power magnetic resonance imaging machines, which are the most important commercial application of the phenomenon to this day. And scientists use superconductors in advanced experiments every day. For instance, particle accelerators at the Large Hadron Collider in Geneva rely on superconducting coils to generate magnetic fields that steer and focus beams of protons. Some of the most accurate measurements in all of science are done thanks to superconducting quantum interference devices, or SQUIDs.
And finally, superconducting electrical transmission lines are here. Wires based on high-temperature superconductors (with liquid nitrogen–based cryogenics, which are technically simpler and much cheaper than liquid helium–based ones) have recently become commercially available. A South Korean utility plans to install them on a large scale. Some U.S. scientists now say that it may be easier to get permits for and build a national superconducting supergrid than construct a conventional high-voltage system.
The American Society of Magazine Editors announced the finalists for the National Magazine Awards yesterday. These awards are also called “Ellies” because the winners receive a reproduction of Calder’s sculpture “Elephant.”
Scientific American got two nominations: one for “General Excellence” in the category of Finance, Technology and Lifestyle Magazines (apparently there is no science category); the other one for best single-topic issue, which mentioned our September 2010 issue “The End”. (The special issue is available as an iPad app, in a single package called “Origins and Endings,” which also includes our September 2009 Origins special issue.)
Congratulations to the entire SciAm staff, and especially to Editor-in-Chief Mariette DiChristina who has shepherded the magazine through a (now we can definitely say) successful relaunch last year. Kudos to Roger Black’s studio for the new design and to our own Design Director Mike Mrak and his staff for making the magazine beautiful month after month. My colleague Michael Moyer deserves special credit for editing the “End” special issue, which included his insightful essay. That issue also included my colleague George Musser’s incredible feature article on whether time could end.
The winners will be announced on May 9.
Over the years, SciAm has earned 16 Ellie nominations and 5 Ellie awards. In addition, in 1999 former Editor-in-Chief was inducted into ASME’s Magazine Editors’ Wall of Fame.
This video is no April’s fool joke: the Earth really is shaped like a potato. Only that the shape you see here is, hum, slightly exaggerated to highlight its irregularities. Another caveat: what is depicted here is not the shape of the planet, but rather the shape of an idealized sea-level surface extending around the entire globe — a surface that Earth scientists call the geoid.
The video is the most accurate reconstruction of the geoid to date, and was released by the European Space Agency at a scientific meeting in Munich on March 31 (see video of press conference here). It is based on data collected by ESA’s Gravity Field and Steady-State Ocean Circulation Explorer (GOCE). The five-meter long, arrowhead-shaped probe has been in a low orbit for almost exactly two years, taking painstakingly accurate measurements of the gravitational field of Earth.
Alas, many of the web sites and blogs that reported on GOCE’s results yesterday got it wrong. Contrary to what was stated in news stories and in blogs, the geoid is not a surface on which the strength of gravity is the same everywhere.
The geoid is, in the words of an oceanographer on the GOCE team, a surface such that if you placed a marble anywhere on it, it would stay there rather than rolling in any direction. Another way of saying it is, imagine you were an engineer traveling around the world with a spirit level. Then wherever you go, the level would be exactly parallel to the geoid at that place. Yet another equivalent definition: it is a surface that’s everywhere perpendicular to the direction of a plumb-bob, or in other words, to the gravitational field.
Gravity need not have the same strength everywhere on the geoid. In other words, if you could walk on the geoid you would see gravity always pointing exactly downwards, but your weight could slightly change from one region to another.
The misunderstanding may have stemmed from the confusion of two concepts from calculus: a function and its derivative. In this case, it’s actually two concepts from multivariable calculus: a vector field and its potential. The potential is the gravitational potential; the vector field is the gravity field. At any point in space, the gravity field represents the direction in which the gravitational potential rises fastest. Its magnitude, or length, is the rate of change of the potential. The field is called the gradient vector field of the potential and it is the 3-D version of a derivative. (To be pedantic: for historical reasons, the gravity field is defined to be the opposite vector to the gradient of the potential.)
If you were to follow a field line of the field – a curve which is at every point tangent to the vector field at that point – you would follow a curve of steepest ascent on the gravitational potential. That’s why it is called the gradient. (Note that because of inertia, the field lines of the gravitational field are not necessarily the trajectories of a body in free fall. The field won’t tell you in which direction you’ll be moving — only in which direction you will accelerate.)
Gradient vector fields are difficult to grasp in 3-D, because picturing the potential (to visualize the ascent or descent) would require a fourth dimension. But a 2-D analogy might help. Think of how water trickles down a hilly surface. Neglecting inertia, a droplet of water would follow a curve of steepest descent, which is a vector field made of vectors along the surface. In this analogy, the elevation of the droplet is the analogue of the potential at that point. And the analogue of the geoid is an elevation contour on a topographical map: the potential – the height — is constant along the contour. Obviously, some parts of this contour can be on steep terrain, whereas others can be on mild gradients. (On a topographical map, the slope is steeper where many different contour lines are crowded together.) The magnitude of the gradient vector field – which represents the steepness of a slope – is not the same at all points of the contour.
Thus, gravity does change in strength along the geoid, and so the geoid is not “a shape where the gravity is the same no matter where you stood on it,” as one blogger put it. Nor is it “how the Earth would look like if its shape were distorted to make gravity the same everywhere on its surface,” as reported by a major news site, which went on to add that “areas of strongest gravity are in yellow and weakest in blue,” apparently not noticing the contradiction with the previous sentence.
There are many reasons why the geoid isn’t a sphere. First and foremost, the Earth itself isn’t a sphere. It is closer to an ellipsoid, being flattened at the poles by the centrifugal force of its own rotation. But the planet isn’t an ellipsoid either, because of topography. Mountains and valleys are asymmetrical distributions of mass. That mass distribution affects the gravitational field and makes the geoid asymmetrical too.
Then there is the Earth’s inner and outer structure. The oceanic crust and the landmasses of the continental shelves are made of different materials that have different densities; and farther down, the mantle is not uniform either, with regions that have slightly different compositions and temperatures, and thus slightly different densities. “The variations,” Reiner Rummel, a senior GOCE scientist, told me in an email, “reflect processes in the deep earth mantle such as descending tectonic plates and hot mantle plumes.”
All of which affects the strength of the gravitational field, and the GOCE’s orbit. The probe measures these effects thus it measures the gravitational field.
The geoid is not to be confused with anomalies due to the topography of the land – although it is affected in part by it. Also – and here is where things get really tricky – it is not to be confused with the actual sea level.
Wait a minute, you say, but isn’t the ocean supposed to be at sea level by definition? Except for relatively small perturbations such as waves or tides, you say, shouldn’t the surface be exactly horizontal (that is, perpendicular to the gravitational field) everywhere? Isn’t that, after all, what “sea level” means? Yes, to some approximation. However, the oceans are not homogenous. Differences of salinity and temperature make them more or less dense. Moreover, the Earth’s rotation produces forces that keep the oceans in constant motion.
Just like the water in a river is not all at the same level – after all, it flows to the sea for a reason – the water in the ocean also has differences in height. And by height, it should be clear by now, I mean elevation relative to the geoid, which can be positive or negative. Oceanographers thus speak of the “topography” of the ocean, meaning that it really has hills and valleys. In fact, here is where the “ocean circulation” part of GOCE’s name comes into play: one of the main goals of the mission is to measure the oceans’ topography and from that deduce the structure of the oceanic currents.
The deviations of the geoid from the simplified, ellipsoidal model of the Earth are substantial: they range from 100 meters below (dark blue in the video) to 80 meters above (yellow), Rummel says.
The surface in the video is the geoid amplified by a factor 7,000, says GOCE mission manager Rune Floberghagen.
"It has often been said there's so much to be read, you never can cram all those words in your head.
So the writer who breeds more words than he needs is making a chore for the reader who reads.
That's why my belief is the briefer the brief is, the greater the sigh of the reader's relief is."
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