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tgb comments on Fermi Estimates - Less Wrong

51 Post author: lukeprog 11 April 2013 05:52PM

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Comment author: tgb 07 April 2013 01:49:41PM 9 points [-]

I had the pleasure the other day of trying my hand on a slightly unusual use of Fermi estimates: trying to guess whether something unlikely has ever happened. In particular, the question was "Has anyone ever been killed by a falling piano as in the cartoon trope?" Others nearby at the time objected, "but you don't know anything about this!" which I found amusing because of course I know quite a lot about pianos, things falling, how people can be killed by things falling, etc. so how could I possibly not know anything about pianos falling and killing people? Unfortunately, our estimate gave it at around 1-10 deaths by piano-falling so we weren't able to make a strong conclusion either way over whether this happened. I would be interested to hear if anyone got a significantly different result. (We only considered falling grands or baby grands to count as upright pianos, keyboards, etc. just aren't humorous enough for the cartoon trope.)

Comment author: gwern 07 April 2013 11:51:59PM 18 points [-]

I'll try. Let's see, grands and baby grands date back to something like the 1700s; I'm sure I've heard of Mozart or Beethoven using pianos, so that gives me a time-window of 300 years for falling pianos to kill people in Europe or America.

What were their total population? Well, Europe+America right now is, I think, something like 700m people; I'd guess back in the 1700s, it was more like... 50m feels like a decent guess. How many people in total? A decent approximation to exponential population growth is to simply use the average of 700m and 50m, which is 325, times 300 years, 112500m person-years, and a lifespan of 70 years, so 1607m persons over those 300 years.

How many people have pianos? Visiting families, I rarely see pianos; maybe 1 in 10 had a piano at any point. If families average a size of 4 and 1 in 10 families has a piano, then we convert our total population number to, (1607m / 4) / 10, 40m pianos over that entire period.

But wait, this is for falling pianos, not all pianos; presumably a falling piano must be at least on a second story. If it simply crushes a mover's foot while on the porch, that's not very comedic at all. We want genuine verticality, real free fall. So our piano must be on a second or higher story. Why would anyone put a piano, baby or grand, that high? Unless they had to, that is - because they live in a city where they can't afford a ground-level apartment or house.

So we'll ask instead for urban families with pianos, on a second or higher story. The current urban percentage of the population is hitting majority (50%) in some countries, but in the 1700s it would've been close to 0%. Average again: 50+0/2=25%, so we cut 40m by 75% to 30m. Every building has a ground floor, but not every building has more than 1 floor, so some urban families will be able to live on the ground floor and put their piano there and not fear a humorously musical death from above. I'd guess (and here I have no good figures to justify it) that the average urban building over time has closer to an average of 2 floors than more or less, since structural steel is so recent, so we'll cut 30m to 15m.

So, there were 15m families in urban areas on non-ground-floors with pianos. And how would pianos get to non-ground-floors...? By lifting, of course, on cranes and things. (Yes, even in the 1700s. One aspect of Amsterdam that struck me when I was visiting in 2005 was that each of the narrow building fronts had big hooks at their peaks; I was told this was for hoisting things up. Like pianos, I shouldn't wonder.) Each piano has to be lifted up, and, sad to say, taken down at some point. Even pianos don't live forever. So that's 30m hoistings and lowerings, each of which could be hilariously fatal, an average of 0.1m a year.

How do we go from 30m crane operations to how many times a piano falls and then also kills someone? A piano is seriously heavy, so one would expect the failure rate to be nontrivial, but at the same time, the crews ought to know this and be experienced at moving heavy stuff; offhand, I've never heard of falling pianos.

At this point I cheated and look at the OSHA workplace fatalities data: 4609 for 2011. At a guess, half the USA population is gainfully employed, so 4700 out of 150m died. Let's assume that 'piano moving' is not nearly as risky as it sounds and merely has the average American risk of dying on the job.

We have 100000 piano hoistings a year, per previous. If a team of 3 can do lifts or hoisting of pianos a day, then we need 136 teams or 410 people. How many of these 410 will die each year, times 300? (410 * (4700/150000000))*300 = 3.9

So irritatingly, I'm not that sure that I can show that anyone has died by falling piano, even though I really expect that people have. Time to check in Google.

Searching for killed by falling piano, I see:

But no actual cases of pianos falling a story onto someone. So, the calculation may be right - 0 is within an order of magnitude of 3.9, after all.

Comment author: Qiaochu_Yuan 08 April 2013 12:30:08AM *  19 points [-]

0 is within an order of magnitude of 3.9, after all.

No it's not! Actually it's infinitely many orders of magnitude away!

Comment author: NancyLebovitz 10 April 2013 05:59:04PM 5 points [-]

Nitpick alert: I believe pianos used to be a lot more common. There was a time when they were a major source of at-home music. On the other hand, the population was much smaller then, so maybe the effects cancel out.

Comment author: gwern 10 April 2013 06:08:39PM 1 point [-]

I wonder. Pianos are still really expensive. They're very bulky, need skilled maintenance and tuning, use special high-tension wires, and so on. Even if technological progress, outsourcing manufacture to China etc haven't reduced the real price of pianos, the world is also much wealthier now and more able to afford buying pianos. Another issue is the growth of the piano as the standard Prestigious Instrument for the college arms races (vastly more of the population goes to college now than in 1900) or signaling high culture or modernity (in the case of East Asia); how many pianos do you suppose there are scattered now across the USA compared to 1800? Or in Japan and China and South Korea compared to 1900?

And on the other side, people used to make music at home, yes - but for that there are many cheaper, more portable, more durable alternatives, such as cut-down versions of pianos.

Comment author: RichardKennaway 10 April 2013 08:25:28PM 4 points [-]

Pianos are still really expensive.

Concert grands, yes, but who has room for one of those? Try selling an old upright piano when clearing a deceased relative's estate. In the UK, you're more likely to have to pay someone to take it away, and it will just go to a scrapheap. Of course, that's present day, and one reason no-one wants an old piano is that you can get a better electronic one new for a few hundred pounds.

But back in Victorian times, as Nancy says elsethread, a piano was a standard feature of a Victorian parlor, and that went further down the social scale that you are imagining, and lasted at least through the first half of the twentieth century. Even better-off working people might have one, though not the factory drudges living in slums. It may have been different in the US though.

Comment author: gwern 10 April 2013 08:53:55PM 3 points [-]

Concert grands, yes, but who has room for one of those? Try selling an old upright piano when clearing a deceased relative's estate.

Certainly: http://www.nytimes.com/2012/07/30/arts/music/for-more-pianos-last-note-is-thud-in-the-dump.html?_r=2&ref=arts But like diamonds (I have been told that you cannot resell a diamond for anywhere near what you paid for it), and perhaps for similar reasons, I don't think that matters to the production and sale of new ones. That article supports some of my claims about the glut of modern pianos and falls in price, and hence the claim that there may be unusually many pianos around now than in earlier centuries:

With thousands of moving parts, pianos are expensive to repair, requiring long hours of labor by skilled technicians whose numbers are diminishing. Excellent digital pianos and portable keyboards can cost as little as several hundred dollars. Low-end imported pianos have improved remarkably in quality and can be had for under $3,000. “Instead of spending hundreds or thousands to repair an old piano, you can buy a new one made in China that’s just as good, or you can buy a digital one that doesn’t need tuning and has all kinds of bells and whistles,” said Larry Fine, the editor and publisher of Acoustic & Digital Piano Buyer, the industry bible.

At least, if we're comparing against the 1700s/1800s, since the article then goes on to give sales figures:

So from 1900 to 1930, the golden age of piano making, American factories churned out millions of them. Nearly 365,000 were sold at the peak, in 1910, according to the National Piano Manufacturers Association. (In 2011, 41,000 were sold, along with 120,000 digital pianos and 1.1 million keyboards, according to Music Trades magazine.)

(Queen Victoria died in 1901, so if this golden age 1900-1930 also populated parlors, it would be more accurate to call it an 'Edwardian parlor'.)

Comment author: OrphanWilde 16 April 2013 12:08:39AM 2 points [-]

We got ~$75 for one we picked up out of somebody garbage in a garage sale, and given the high interest we had in it, probably could have gotten twice that. (Had an exchange student living with us who loved playing the piano, and when we saw it, we had to get it - it actually played pretty well, too, only three of the chords needed replacement. It was an experience loading that thing into a pickup truck without any equipment. Used a trash length of garden hose as rope and a -lot- of brute strength.)

Comment author: NancyLebovitz 10 April 2013 06:56:24PM *  2 points [-]

I was basing my notion on having heard that a piano was a standard feature of a Victorian parlor. The original statement of the problem just specifies a piano, though I grant that the cartoon version requires a grand or baby grand. An upright piano just wouldn't be as funny.

These days, there isn't any musical instrument which is a standard feature in the same way. Instead, being able to play recorded music is the standard.

Thanks for the link about the lack of new musical instruments. I've been thinking for a while that stability of the classical orchestra meant there was something wrong, but it hadn't occurred to me that we've got the same stability in pop music.

Comment author: gwern 10 April 2013 07:04:12PM 1 point [-]

I was basing my notion on having heard that a piano was a standard feature of a Victorian parlor.

Sure, but think how small a fraction of the population that was. Most of Victorian England was, well, poor; coal miners or factory workers working 16 hour days, that sort of thing. Not wealthy bourgeoisie with parlors hosting the sort of high society ladies who were raised learning how to play piano, sketch, and faint in the arms of suitors.

An upright piano just wouldn't be as funny.

Unless it's set in a saloon! But given the low population density of the Old West, this is a relatively small error.

Comment author: David_Gerard 11 April 2013 06:20:10PM *  0 points [-]

That article treats all forms of synthesis as one instrument. This is IMO not an accurate model. The explosion of electronic pop in the '80s was because the technology was on the upward slope of the logistic curve, and new stuff was becoming available on a regular basis for artists to gleefully seize upon. But even now, there's stuff you can do in 2013 that was largely out of reach, if not unknown, in 2000.

Comment author: satt 12 April 2013 01:38:38AM 2 points [-]

But even now, there's stuff you can do in 2013 that was largely out of reach, if not unknown, in 2000.

Have any handy examples? I find that a bit surprising (although it's a dead cert that you know more about pop music than I do, so you're probably right).

Comment author: David_Gerard 12 April 2013 06:54:26AM *  1 point [-]

I'm talking mostly about new things you can do musically due to technology. The particular example I was thinking of was autotune, but that was actually invented in the late 1990s (whoops).

But digital signal processing in general has benefited hugely in Moore's Law, and the ease afforded by being able to apply tens or hundreds of filters in real time. The phase change moment was when a musician could do this in faster than 1x time on a home PC. The past decade has been mostly on the top of the S-curve, though.

Nevertheless, treating all synthesis as one thing is simply an incorrect model.

Comment author: satt 13 April 2013 01:50:20PM 2 points [-]

Funny coincidence. About a week ago I was telling someone that people sometimes give autotune as an example of a qualitatively new musical/aural device, even though Godley & Creme basically did it 30+ years ago. (Which doesn't contradict what you're saying; just because it was possible to mimic autotune in 1979 doesn't mean it was trivial, accessible, or doable in real time. Although autotune isn't new, being able to autotune on an industrial scale presumably is, 'cause of Moore's law.)

Comment author: zerker2000 17 April 2013 03:14:03AM *  1 point [-]

Granular synthesis is pretty fun.

Comment author: satt 17 April 2013 10:51:09PM 0 points [-]

Agreed, although I don't know how impractical or unknown it was in 2000 — I remember playing with GranuLab on my home PC around 2001.

Comment author: Larks 08 April 2013 02:05:29AM 3 points [-]

Average again: 50+0/2=25%, so we cut 40m by 75% to 30m.

To 10m, surely?

Comment author: private_messaging 10 April 2013 08:06:44PM *  2 points [-]

The workplace fatalities really gone down recently, with all the safe jobs of sitting in front of the computer. You should look for workplace fatalities in construction, preferably historical (before safety guidelines). Accounting for that would raise the estimate.

A much bigger issue is that one has to actually stand under the piano as it is being lifted/lowered. The rate of such happening can be much (orders of magnitude) below that of fatal workplace accidents in general, and accounting for this would lower the estimate.

Comment author: gwern 10 April 2013 08:18:10PM 1 point [-]

You should look for workplace fatalities in construction, preferably historical (before safety guidelines).

I don't know where I would find them, and I'd guess that any reliable figures would be very recent: OSHA wasn't even founded until the 1970s, by which point there's already been huge shifts towards safer jobs.

A much bigger issue is that one has to actually stand under the piano as it is being lifted/lowered. The rate of such happening can be much (orders of magnitude) below that of fatal workplace accidents in general, and accounting for this would lower the estimate.

That was the point of going for lifetime risks, to avoid having to directly estimate per-lifting fatality rates - I thought about it for a while, but I couldn't see any remotely reasonable way to estimate how many pianos would fall and how often people would be near enough to be hit by it (which I could then estimate against number of pianos ever lifted to pull out a fatality rate, so instead I reversed the procedure and went with an overall fatality rate across all jobs).

Comment author: Decius 18 April 2013 04:38:39AM 0 points [-]

You also need to account for the fact that some proportion of piano-hoister work-related fatalities will be to other factors like heatstroke or heart attack or wrapping the rope around their arm.

Comment author: Kindly 08 April 2013 02:36:33PM *  2 points [-]

To a very good first approximation, the distribution of falling piano deaths is Poisson. So if the expected number of deaths is in the range [0.39, 39], then the probability that no one has died of a falling piano is in the range [1e-17, 0.677] which would lead us to believe that with a probability of at least 1/3 such a death has occurred. (If 3.9 were the true average, then there's only a 2% chance of no such deaths.)

Comment author: gwern 08 April 2013 03:09:56PM 6 points [-]

I disagree that the lower bound is 0; the right range is [-39,39]. Because after all, a falling piano can kill negative people: if a piano had fallen on Adolf Hitler in 1929, then it would have killed -5,999,999 people!

Comment author: Kindly 08 April 2013 10:31:22PM 2 points [-]

Sorry. The probability is in the range [1e-17, 1e17].

Comment author: Qiaochu_Yuan 09 April 2013 05:17:54AM 3 points [-]

That is a large probability.

Comment author: Decius 18 April 2013 04:43:21AM 1 point [-]

It's for when you need to be a thousand million billion percent sure of something.

Comment author: Mark_Eichenlaub 09 April 2013 06:16:21AM *  1 point [-]

A decent approximation to exponential population growth is to simply use the average of 700m and 50m

That approximation looks like this

It'll overestimate by a lot if you do it over longer time periods. e.g. it overestimates this average by about 50% (your estimate actually gives 375, not 325), but if you went from 1m to 700m it would overestimate by a factor of about 3.

A pretty-easy way to estimate total population under exponential growth is just current population * 1/e lifetime. From your numbers, the population multiplies by e^2.5 in 300 years, so 120 years to multiply by e. That's two lifetimes, so the total number of lives is 700m*2. For a smidgen more work you can get the "real" answer by doing 700m * 2 - 50m * 2.

Comment author: mwengler 26 April 2013 05:58:49PM 6 points [-]

Cecil Adams tackled this one. Although he could find no documented cases of people being killed by a falling piano (or a falling safe), he did find one case of a guy being killed by a RISING piano while having sex with his girlfriend on it. What would you have estimated for the probability of that?

From the webpage:

The exception was the case of strip-club bouncer Jimmy Ferrozzo. In 1983 Jimmy and his dancer girlfriend were having sex on top of a piano that was rigged so it could be raised or lowered for performances. Apparently in the heat of passion the couple accidentally hit the up switch, whereupon the piano rose and crushed Jimmy to death against the ceiling. The girlfriend was pinned underneath him for hours but survived. I acknowledge this isn’t a scenario you want depicted in detail on the Saturday morning cartoons; my point is that death due to vertical piano movement has a basis in fact.

Comment author: [deleted] 27 April 2013 10:22:26AM 1 point [-]

Reality really is stranger than fiction, is it.