## Archive for the ‘Math Mistakes’ Category

### Failure and Bottled Water

June 2, 2009

It’s no secret that we love Fail Blog (G-rated version here), because they have posts like this:

And this:

(Although this sign really makes me love New Cuyama.  Who else puts random addition on their City Sign?    New Cuyama is about 40 miles due north of Santa Barbara, and I totally want to visit there now.)

Fail Blog also posts signs like this:

This last post actually made me pause for a while.  Yes, the 167 bottles was clearly wrong, but working backwards suggests that this sign intended to say that Americans drank 50 billion bottles of water in 2006, that 32% (16 billion) were recycled, and the remaining 68% (34 billion) ended up in the landfill.  And that just seemed wrong.  There are about 300 million people in the United States [assuming that Americans refers to the single country and not all of North or South America], so that would means that in 2006 there were over 160 bottles of water sold per person.  And that couldn’t be right, could it?

Actually, it could.  According to the Beverage Marketing Company, there were over 8.7 billion gallons of bottled water sold in the US in 2007.   If there are 16 oz. of water per bottle, then each gallon would give 8 bottles, leading to well over 60 billion bottles of water sold in 2007, which is in the same ballpark as above.  Who knew?

(This reminds me of Chris Jordan’s photographs.  And lo, it turns out he has one of bottles, partway down the page.)

### Math Teachers at Play #7 plus a video

May 15, 2009

Math Teachers at Play #7 is living this week at Homeschool Bytes, with lots of neat posts (like this one about Math and Language Reversals).

Incidentally, if you haven’t seen it yet, on April 30 The Daily Show had a great piece [not quite safe for work in places] about the Large Hadron Collider (currently due to start up again in the fall).   Pay careful attention to the calculation that the world had a 50% chance of being destroyed; Walter probably checks this site regularly.  (There’s a lot more discussion at Good Math, Bad Math.)

[Hmmm, the embedding isn't working.  Bummer.  You'll have to go here.]

### Sunscreen confusion

May 14, 2009

An article in the New York Times describes consumer confusion over the ever-rising SPF numbers (used to rate the efficacy of sunscreen lotions), and their interpretation.

Unfortunately, the NYT adds to the confusion with the following:

The difference in UVB protection between an SPF 100 and SPF 50 is marginal. Far from offering double the blockage, SPF 100 blocks 99 percent of UVB rays, while SPF 50 blocks 98 percent. (SPF 30, that old-timer, holds its own, deflecting 96.7 percent).

Technically they’re right:  doubling the blockage is not the same as halving your radiation exposure.  But in terms of safety, the issue isn’t how much UV exposure you’ve avoided, but rather how much UV actually gets to your skin cells (which would then be a 2% versus 1% comparison).

According to the article, SPF measures how much longer a person wearing sunscreen can be exposed to sunlight before getting a burn, when compared to someone wearing no sunscreen.  Someone wearing SPF 50 can remain in the sun 50 times longer than someone with no sunscreen, and so SPF 100 sunscreen provides the wearer with twice the protection (in terms of time) as SPF 50 sunscreen.

It turns out there is a sense in which SPF100 is not twice as effective as SPF50 in protecting your skin, but it has nothing to do with the 99%/98% comparison.

According to the NY Times, “a multiyear randomized study of about 1,600 residents of Queensland, Australia” found that most users applied at most half of the recommended amount of sunscreen.

“If people are putting on about half, they are receiving half the protection,” said Yohini Appa, the senior director of scientific affairs at Johnson & Johnson, of which Neutrogena is a subsidiary.

But in fact they are receiving far less than half the protection:   a 2007  British Journal of Dermatology study noted that cutting the amount of sunscreen in half did not reduce the effective SPF in half, but rather reduced it geometrically to its square root.

If a person uses half of the recommended amount of an SPF50 sunscreen, they’ll get the protection of an SPF7 (since 7.1 is roughly √50), while similarly underapplying SPF100 sunscreen gets the protection of SPF10.

Apparently, if you’re looking for the protection of an SPF30 product, but like most people tend to under-apply sunscreen, you should be shopping for sunscreen rated as SPF 900.   No word yet on when such products will hit the marketplace.

(One wonders: does this work the other way ’round?  If I apply TWICE the recommended amount of a cheaper SPF8 sunscreen, do I end up with the protection of SPF64 sunscreen?)

### Counting Trouble

May 12, 2009

There have been posts here before about how bees and elephants can count small numbers.  Sometimes it’s a little difficult for us humans, however.

see more pwn and owned pictures

### Unit Comparison

May 7, 2009

Here’s the price of Flax Oil at our local Wegmans:

Is that too hard to read?  The bottle on the far left is 16 fluid ounces for $15.49, and the bottle next to it is 8 fluid ounces for$8.49.  As it happens, even though the bottles are different sizes it’s pretty easy to compare the prices for this particular example:  the large bottle is just under $1/ounce and the small bottle just over, so the prices are similar but the large bottle is the better deal. In general, though, it can be difficult to compare different sizes of the same product so the store provides a unit comparison to the left of the price. In theory this gives the price per ounce/pound/quart or whatever makes sense. The most important part of this is that you use the same unit for the different brands/sizes — that’s what makes for the comparison. Yet if you look at this picture [which is a little grainy, because Wegmans doesn't like people taking photos so this was done on the sly with a cell phone] you can see that the small bottle is$33.96 per quart, while the larger bottle is only $15.49 per… pint. Every other bottle on that shelf has the price per quart except for that bottle of Flax Oil, which happens to be the Store’s Name Brand. What makes it worse is that when TwoPi first noticed this a couple weeks earlier, Wegmans Flax Oil was more expensive and so the large bottle was not the better deal. That make the sneaky “per pint” price even worse, because it hid the truth. Wegmans, we love your store but in this particular regard, you blew it. ### Math in the Movies May 6, 2009 A recently ran across a great resource page of “Math in the Movies” from MathBits. This page lists clips of movies (with links if they’re available online) and worksheets that can be used in the classroom. Two of the examples relate to Star Trek, which as you all know (right?) will be in theaters starting this weekend. In Episode 20 (“Court Martial”) of The Original Series, James T. Kirk is accused of murdering Benjamin Finney, the Records Officer. At one point late into the episode, Kirk uses the computer to hear the heartbeats of everyone on The Enterprise. As he explains: Gentlemen, this computer has an auditory sensor. It can, in effect, hear sounds. By installing a booster we can increase that capability on the order of one to the fourth power. The computer should be able to bring us every sound occurring on the ship. One to the fourth power? Not so impressive (and was I the only one thinking that you’d hear a lot more than heartbeats if every sound was magnified? Wouldn’t breathing and moving be really LOUD? But I’d better be careful with my critiques, because I’m a big fan of The Next Generation and a recent viewing revealed that it isn’t immune to problems either.) The MathBits worksheet to accompany the scene is here. They don’t link to a video clip, but you can see it on YouTube. Actually, you can see the entire episode on YouTube: the mistake starts at 38:10. (Edit 5/9: the embedding no longer works, but you can see the episode at this link.) ### Faulty Unit Conversion April 16, 2009 Language Log has a post up (via HeadsUp the blog) about a Fox News story that had some metric issues: The tests involved head-on crashes between the fortwo and a 2009 Mercedes C Class, the Fit and a 2009 Honda Accord and the Yaris and the 2009 Toyota Camry. The tests were conducted at 40 miles per hour (17 kilometers per liter), representing a severe crash. The comments are especially entertaining. On a personal note, I have a 1999 Camry that does 23.4 km/l highway, 12.8 km/l city. ### We sleep HOW MUCH? March 31, 2009 I’m an alum of UW-Madison, and have many fond memories of the place (indeed, TwoPi and I met in grad school there). As a bonus, they have a good alumni magazine. But even the best magazines make mistakes from time to time, and the Spring 2009 issue of On Wisconsin includes a graph that is simply wrong. The graph, illustrating how much time we sleep compared to other activities, appeared in the article “Bedtime Story” (description “Why do we snooze? UW researchers are pursuing an unconventional theory — that our brains downsize while we sleep, getting us ready to face another day.”) Here’s what the graph looks like: Take a look at the two biggest circles: we spend 36% of our life sleeping, and 19% on other [unlisted] activities. But that gray circle is pretty small — you could fit 3 or 4 of them into the Sleeping circle. I grabbed a ruler and, sure enough, its the diameters, not the areas, that have the 36:19 ratio. The distinction is even more striking when you look at the tiny circles: we spend 36 times as much of our time sleeping as we do on telephone calls, mail, and email and that’s a LOT, but it’s not a ratio of 1296 (as the areas would suggest). So I created a graph in which the areas are in correct proportion: There’s still a clear difference in how much time we spend doing different things, but it no longer looks like 61% of our lives are spent in bed. ### Tidal force, or The Moon and the Mosquito revisited March 30, 2009 As Ξ noted in an earlier post, the claim that the gravitational pull of a mosquito is stronger than the gravitational pull of the moon is off by a fair bit … roughly five orders of magnitude. The original author of the claim, George Abell, was an astronomer. Aren’t astronomers supposed to be good at working with large numbers? Wondering if he had been misquoted in Scientific American, I set out to find out what I could about this claim. A google search turned up a fair number of sources that describe Abell’s claims; most of their accounts are similar to that of Lilienfeld and Arkowitz. One of the more widely read accounts appears in The Skeptics Dictionary (by Robert T. Carroll): Astronomer George O. Abell claims that a mosquito would exert more gravitational pull on your arm than the moon would (Abell 1979). The secondary literature is pretty much in agreement: Abell claimed that the mosquito exerts a stronger gravitational pull than the moon. (And clearly that claim is false.) A quick stop at the library of a nearby college turned up Abell’s original article. He had written a piece for the Skeptical Inquirer, a review of a book on the putative effects of the moon on human behavior. The book’s author had suggested a plausible mechanism for such influence: the body is largely made up of water, and we all know the moon is a primary cause of tides on the Earth. Abell’s discussion notes the source of the Moon’s influence on terrestrial tides: not the gravitational pull of the moon, but rather the difference in that force between the nearest and farthest points on the Earth. Because of that difference, the Earth is (very slightly) distorted, with its fluid surface in motion attempting to achieve equilibrium. Abell notes that while the Sun’s gravitational pull on the Earth is more than 100 times stronger than the Moon’s, its tidal force — the difference in the Sun’s pull over the diameter of the Earth — is less than half that of the tidal force of the Moon. If the Moon’s influence on human behavior were tidal (acting on the fluids in the body), then that tidal effect would be due to differences in the Moon’s gravitational pull on the fluid in different parts of your body — differences due to the fact that (for a 6 foot tall person) part of their body could be six feet closer to the center of mass of the Moon than other parts of their body. Abell quotes this difference as being “about one part in $3 \times 10^{13}$ (or 30 trillion) of the weight of that fluid”, and states that the copy of a magazine in the reader’s hand is tens of thousands of times more significant in producing tidal forces. If we take the weight of the magazine to be roughly 1/2 lb [based on weighing a stack of AMS Notices], or 2.2 Newtons, leading to a mass of 0.2 kg, then the tidal force due to the magazine (over distances ranging from .2 m to .5 m — assuming I hold the book about 20 cm from my torso, and my torso is roughly 30 cm deep) will be $\frac{G\cdot M \cdot (0.2)}{.2^2} - \frac{G \cdot M \cdot (0.2)}{.5^2 }$ (where G is the gravitational constant, and M is the mass of the fluids in your body), which works out to approximately $2.8 \times 10^{-10} \cdot M$ newtons. A similar calculation for the lunar tidal force acting on my body when I am standing (taking my height to be 2 meters): $\frac{G\cdot M \cdot (7.3483 \times 10^{22})}{(3.844\times 10^8 - 2)^2} - \frac{G\cdot M \cdot (7.3483 \times 10^{22})}{(3.844\times 10^8)^2}$ which is roughly $3.5 \times 10^{-13} \cdot M$ newtons. So indeed, as Abell claimed, it is reasonable to conclude that a small book or magazine in your hand exerts a tidal force on the order of thousands of times stronger than the moon on your body. But wait! Where are the mosquitoes? Abell described a second potential source of lunar influence on the human body. He writes: What might matter is the difference between your weight in the presence of the moon’s gravitational effect and what it would be if there were no moon. At the most, that difference amounts to only 0.01 gram, or about 0.0003 ounces, less than the effect of a mosquito on your shoulder. In the paragraph leading up to that passage, Abell noted that this isn’t the same as the gravitational pull of the Moon on your body, since the Moon is also pulling on the Earth (so you have a slightly larger acceleration toward the Moon than the center of the Earth does — in effect, this is the tidal force calculation in a different guise). It isn’t quite clear what Abell means by “the effect of a mosquito on your shoulder”: is he referring to a gravitational (tidal) effect due to the mosquito, or is he just referring to the weight of the mosquito — the amount by which it increases your weight when it lands? I’ll leave the analysis of these two interpretations as exercises for our readers. But the conclusions I found: if we compute the tidal force of the mosquito on your body, it is far greater than the tidal force of the moon on your body (because of its proximity versus the great distance to the Moon). If instead we believe that Abell is claiming that the weight (mass) of the mosquito is greater than the 0.01 g that he had computed, then he is wrong: Ξ had generously estimated the mass of a mosquito at 5 mg; Abell’s estimate of the Moon’s influence on a person’s weight is about twice that. Abell’s review of Arnold Leiber’s The Lunar Effect: Biological Tides and Human Emotions (1st ed, 1978) appeared in the Skeptical Inquirer 3 (1979) pp 68 – 73. ### FutureGen March 24, 2009 It’s hard to estimate how much it will cost to start/change a company, especially because it’s reasonable to expect that$1 today is worth less than $1 in the future. Do you use today’s prices, or take inflation into account? On the other hand, if what you’re doing is comparing costs, it doesn’t really matter which method you use as long as you’re consistent. A mistake with that last bit ended up possibly costing FutureGen its future. Here’s some background. According to the US Department of Energy, “FutureGen is an initiative to equip multiple new clean coal power plants with advanced carbon capture and storage (CCS) technology.” This was announced in 2003, but a year ago the top folk in the Department of Energy decided that it was going to be too expensive to build. Certainly it would have been expensive: over a billion dollars (about 8% of which would have been paid for by China and India as research into cleaner energy that they might be able to use). But not quite as expensive as they thought. According to Scientific American, the Department of Energy (DOE) had essentially forgotten to account for inflation when estimating FutureGen’s projected costs. Specifically, the department had said in 2004 that it would cost$950 million to build, a sum that it last year said had ballooned to $1.8 billion when projected through 2017. In fact, the GAO says, the actual cost considering inflation would be closer to$1.3 billion….

These new figures were released in a 54-page report by the Government Accountability Office (GAO) on March 11, 2009, which you can read here.  In particular, one of the problems was that the Department of Energy was making comparisons of FutureGen as originally planned versus replacing/restructuring, but in one case they were taking inflation into account (that $1 spent in 5 years is worth less than$1 today) and in another case they weren’t, so the numbers weren’t comparable.  In The New York Times, Representative Bart Gordon (a Democrat from Illinois) said,

I am astonished to learn that the top leadership of the Department of Energy in the last administration made critical decisions about our nation’s energy future and capacity to combat global warming based on fundamental budget math errors…This is math illiteracy on a grand scale and with global consequences.

(In the interests of full disclosure, Illinois is the state where FutureGen would be located, so the consequences may have hit closer to home, so to speak.)

With this disclosure and a new administration, FutureGen might be back on the table.  Or maybe not — presumably the price has increased even since those figures were taken into account, and so other comparisons would have to be made before a decison would be made.  Hopefully this time using comparable data.

### The moon versus the mosquito

March 22, 2009

Last month Scientific American reported on whether scientific studies showed an association between the full moon and strange behavior.  The short answer is no, despite a wide-spread belief in the connection.

In the article, Scott O. Lilienfeld and Hal Arkowitz  mention that some people (ancient and current) believe that the moon’s supposed effect might have something to do with its gravitational effect on water, which makes up most of the human body.  But then they add that this effect isn’t strong enough:

As the late astronomer George Abell of the University of California, Los Angeles, noted, a mosquito sitting on our arm exerts a more powerful gravitational pull on us than the moon does.

“Hmmm,” I thought to myself upon reading this, “Is that true?  Is a mosquito really a more significant graviational force than the moon?”

The force of gravity between two objects of mass $M$ and $m$ respectively is given by

$\frac{GMm}{r^2}$

where $G$ is the gravitational constant and $r$ is the radius between the two objects.

In order to compare the force exerted by the moon and a mosquito, $G$ and $M$ (the mass of the individual) are the same in both equations, so we really just need to compare the values of $\frac{m_{moon}}{{\left(r_{moon}\right)}^2}$ and $\frac{m_{bug}}{{\left(r_{bug}\right)}^2}$.

So what are these two amounts?  NASA says that the mass of the moon is 7.3483×1022 kg, which translates to 7.3483×1028 milligrams. The average distance to the earth is 3.844 x 105 km, which is the same as 3.844 x 1011 mm. [I'm using tiny units because we're about to compare this to a mosquito.] So the ratio $\frac{m_{moon}}{{\left(r_{moon}\right)}^2}$ is just about 497,300 mg/mm2.  This amount might be an underestimate, too, because the distance $r_{moon}$ I used was the distance between the moon and earth, but I suspect that is the distance between the centers; we only need to go from the center of mass of the moon to the surface of the earth where our individual is sitting.    But for rough purposes, it will do.

Mosquitoes, on the other hand, appear to weigh 1-5 milligrams each (though some sites say only 1-2 milligrams).  We’ll use the upper bound of 5 milligrams.  For the distance, I feel like we should use the distance between the centers of the mass of the mosquito and individual if we want the overall gravitational effect, but I’m willing to limit the effect  to just a few cells near the mosquito.   A reasonable estimate for the distance from the mosquito’s body to the skin is 1 mm.  The formula $\frac{m_{bug}}{{\left(r_{bug}\right)}^2}$ then becomes only 5 mg/mm2.  And this amount is really an overestimate, since I used the largest possible value for the mass and chose the smallest distance.

So if my estimates here are accurate and I haven’t missed anything significant, the gravitational pull of the moon is on the order of 100,000 times as strong as the gravitational pull of a mosquito;  George Abell — debunker of pseudoscience himself — appears to be quite wrong.

Isn’t that a great photo of the full moon?  Luc Viatour took it in Belgium (© Luc Viatour GFDL/CC). The mosquito comes from the Centers for Disease Control and Protection.

February 5, 2009

I saw this video today on Failblog.  It’s almost 3 minutes long, and has made the rounds — it’s from December 2006 –  but it’s still amusing to listen to the consumer trying repeatedly to explain that yes, there is a difference between $0.002 and 0.002¢, and to further explain to the manager that this was a matter of fact, not a matter of opinion. (The Verizon consumer, George Vaccaro, had been quoted a rate of 0.002¢ per kilobyte while in Canada. He’d used 35893 kilobytes, but was charged$71.79 instead of $0.7179.) According to this Verizon page, they now quote it as “$0.002 per KB or $2.05 per MB” (which are not quite equivalent, but at least not off by two orders of magnitude). You can read the full story here. Verizon officials did eventually admit that the rate of 0.002 cents/KB was incorrect, and his money was refunded. ### Simple Addition: Apparently not so simple January 29, 2009 Poor Montgomery County, Maryland. Someone forgot to check the addition on the estimate of property value in the county, and it turns out that the estimate wasn’t a very good estimate after all. It was off by a whopping$16,000,000,000.  That’s right, sixteen billion dollars — a result of entering $180 billion instead of$164 billion.  To put that kind of money into perspective, it’s….it’s sixteen billion dollars.  There is no way to put that in perspective.   [Except perhaps for  The Washington Post's observation that it is equivalent to the Gross Domestic Product of Jordan.]

Of course, that’s just the value of the property,  not the amount of revenue that the government expected to gain in taxes from the property value, which was used to calculate school budgets.  Ironically, the result was that Montgomery County didn’t get enough money:  money from richer counties is shared with schools in poorer counties, and Montgomery looked extra rich.  The end result is that Montgomery is getting its $24 million, but lawmakers are suddenly trying to find a way to cover the$31 million that was mistakenly given to the other schools from Montgomery’s non-existent tax income.

(“Suddenly?” you ask.  “Didn’t this mistake actually happen in 2007?”  Why yes, yes  it did, in November.  A few people noticed:  one person even sent an email asking if the numbers were correct, but the email was never answered.  Having forgotten to reply to a few emails myself, I can totally see how that happened — you get a question, mean to answer it, get distracted, and all of a sudden it’s fallen down the email queue into The Abyss.  A bit of a problem, nonetheless.  So the matter was tabled, until this past summer when “mid-level number crunchers in state government” (WP) caught it — hooray for the mid-level number crunchers!   Word got back to Montgomery County last month, and this story first hit the airwaves a few weeks ago.)

The moral of the story:  check your math.  And check your email.

### Bochner’s Meditation on the Theorem of Pythagoras

January 28, 2009

The January 2009 issue of The College Mathematics Journal has a Pythagorean theme.  While the articles consist of the usual mix of varied mathematical topics, most of the smaller sidebar inserts contain quotes from books or articles about Pythagoras, and the issue concludes with reviews of recent books by Eli Maor and by Christoph Riedweg on the Pythagorean Theorem and the life of Pythagoras, respectively.

The front cover of the journal has a photograph of a piece created by the artist Mel Bochner, his “response to a visit to a temple in Metapontum”, the city where Pythagoras is said to have died.  Media:  chalk and hazelnuts.  (You can also find a different image of this work, dating to 1972, as the 16th image in the slide show of “Selected Works: 1966 – 2008″ on Bochner’s website.)

I love the simplicity:  illustrating the fundamental ideas of relating the lengths of the sides of a triangle to the areas of the squares on those sides using readily available materials.

I do have one nit to pick, though.  If the intent was to illustrate a right triangle, then the arrangement of hazelnuts is off.  To my eye, the hazelnut grids look exactly like pins on a Geoboard, or lattice points in the plane.  And given that perspective on this image, we see a 2-3-4 triangle, an obtuse triangle, and squares of area 4, 9, and 16.

I suspect that what was intended was something akin to the following:

Here we can view the diameter of each hazelnut as being our unit of length, so that the circular area taken up by each hazelnut suggests the unit of area (the circumscribing square) .

This image differs from Bochner’s piece in a critical way:  Bochner has arranged his hazelnuts with relatively large gaps between each nut, while in the schematic I’ve abutted them to one another, as one would do if the diameter of each nut was a unit of length, and the nuts were being used as a measurement device.

The large amount of space between the nuts is akin to lattice points in the plane, in which it is the gap itself which constitutes the unit of length, and the vertices (or hazelnuts) are our attempt at an approximation to ideal points in the plane.

If the triangle is meant to be a 3-4-5 triangle, the corresponding lattice image would be as follows:

In the end, I find Bochner’s Meditation rather confusing, and to some extent disappointing.

### Proceed with Caution

December 2, 2008

I ran across an interesting math mistake in a post by Bob Murphy on the blog Crash Landing. He quotes from Peter L. Bernstein’s 1993 book Capital Ideas (which is apparently “A savvy appreciation of how a small band of disinterested academics has revolutionized the way Wall Street and its offshore counterparts manage the world’s investment wealth,” according to Kirkus Review).

Anyway, here’s the quote:

[Cowles] must have been a fiendish bridge player. Here is one passage from his notes on the game:

If each of 50 million bridge players in the US plays 200 sessions of 40 deals each, this adds up to 50 million*200*40 = 400 billion hands dealt each in US (sic). The probabilities on any given hand being dealt with 13 cards of one suit are .00000000000156. The chances of a hand with 13 cards of one suit being dealt in the US in any given year, therefore are 400 billion times .00000000000156=.624.

The challenge posed was to find the math mistake. I’m pretty sure I know which one he was referring to, but I’ll share my train of thought anyway.

My first guess was that it has to do with the 400 billion deals going on in the US each year. But in retrospect, I think this number is correct given the assumptions [and assuming that it's only the dealer's hand that's being looked at; there are four people playing with each deal]. What I disagree with is that those assumptions are reasonable. Are there really 50 million bridge players in the US? Maybe. But the US Census estimates that there are only 301,621,157 people in the US as of July 1, 2007; that means that 1 in 6 people is a bridge player. It might be true that 1/6 of the folk in the US know how to play bridge (maybe), but I’m pretty sure that it is not the case that 1/6 of the entire US population deals 8000 hands per year. That’s almost 22 hands per day (and that only counts when that person is the dealer!), each and every day, for an entire year. If that were the case, who would have time for blogging?

So that 400 billion is way off, but because of unreasonable assumptions rather than an actual mistake (and not having read the actual book, it’s possible he knew that he was wildly overestimating the number of hands dealt). Then I turned my attention to the line “The probabilities on any given hand being dealt with 13 cards of one suit are .00000000000156.” This number comes from the fact that the number of ways to choose 13 cards from a deck of 52 is (52 choose 13), or approximately 6.35×1011. There’s only one way to get all 13 cards in a particular suit (say, spades) so the chances of getting that or any particular configuration of cards is 1/(52 choose 13) or approximately 1.57×10-12. So that number fits the paragraph above, if you interpret “a suit” to mean a particular suit. If you just want all the cards to be of the same suit, then you’d have to multiply that probability by 4.

Which means that probability is probably wrong too, but it does depend on interpretation. Finally I turned my attention to last line, about multiplying the probability by 400 billion to find the likelihood of getting all cards in the same suit because there are 400 billion hands dealt. And this, I am certain, is the error that Bob Murphy was referring to. As Bob’s brother pointed out, if they had used a population that was large enough (or indeed, simply multiplied the tiny probability above by 4 to take into account that there are 4 different suits in the deck), the probability of getting all 13 cards in the same suit would appear to be more than one.

The correct way to solve this would be to find the likelihood that a particular hand dealt was not all of a given suit (1-1.57×10-12), raise that to the 400 billion power to find the probability that none of those 400 billion hands dealt were all of a suit, and then subtracting that from 1. Using Excel, it appears that there’s a 99.996% change that someone, somewhere got that particular configuration of cards.

What a rich little paragraph! It looks like Bernstein has written a sequel to the book in 2005; let’s hope that in this version, the math was checked a bit more carefully.