These sorts of problems also come up a lot in real life engineering, when you’re dealing with a client who is brainstorming for ideas on something, and you have to sit there and give them answers to seemingly impossible and inane questions. Such as “how much trash is available to burn at my power plant within 200 miles of Colorado Springs”?
It helps keep the brainstorming at a sane level, and puts a brake on inaccurate speculation.
For example, to the suggestion that we extrapolate to 250 giraffes, we should now find out. The real answer would tell us something about the assumptions we made. For example, is # of giraffes dependent on total population, or on the number of major cities, or on the propensity of one society’s cities to have more zoos than the other?
Also, order of magnitude is power of 10. If we fermi-answer 250 giraffes in the US, and it’s actually 2500, then we’re off by one order of magnitude. Sometimes the same order of magnitude is about the best you can come up with.
I’ll have to help answer 4 such questions this very night. I regularly participate in a pub quiz with a team. The way the guy formats his quiz is to end each section of the sort of questions you might expect with what he refers to as an “anyone’s guess” or “borderline stupid” question, which is something of this sort. He awards an added 5, 3 and 1 points for the teams that come closest. Of course, you often get arguments by two people on your team, each following their own sequence of approximations yielding answers separated an order of magnitude or more.
We ask similar questions when we interview actuarial graduates. We’re less interested in whether they get the correct answer than in the reasoning they use to justify their answer. At the last lot of interviews it was “how many babies were born in New South Wales last year?”
I tried to estimate the error in a typical fermi problem and fell short and realised this was too circular… But it does suggest my initial guess was pessimistic – within an order of magnitude probably is a reasonable expectation, which is good enough for most purposes.
Let us just say that someone being out by a factor of 100 – if they do the same sort of estimation as we’re doing – is the 5% bad luck rather than making any mistake themselves, which I think may be the false assumption some, eg. interviewers, make, which is what upsets some people.
The American Association of Zoo-keepers (http://www.aazk.org/) says they have members from nearly 250 zoos.
Ueeech, data. Don’t get any of it on me!
This is a much more likely. We’re not interested in your answer (which is likely to be no more accurate than a on-the-spot guess), just in how you arrive at it. A subjective evaluation, certainly, but a valid and interesting method of seeing how smart someone is.
But totally unsuitable for an IQ test (which are dubious things anyway, but that’s another thread…).
But you shouldn’t judge them on the answer anyway, only on how they arrive at it. I’d prefer someone who overestimates the number of zoos, or who makes an innocent mental arithmetic error, over someone who arrives at the correct answer (as far as I can determine what that is, of course) by a route which suggests a lack of basic thinking skills (“given that California and Africa are roughly the same size, they must have the same number of giraffes, so …”).
Even so, an error of two orders of magnitude is pretty large. If 250 is the correct answer, then you’d need to estimate less than 3 or more than 2500 giraffes in the US, to be off by that much. The lower bound of that range is obviously silly. The upper bound is not so obviously absurd, but still pretty tough to reach from reasonable assumptions. So I do expect that most people will end up within a factor of 10 from the OP’s guess, and that it takes more than just bad luck to get the answer really spectacularly wrong.
I was once grading a homework consisting of Fermi problems. One of them: “A paleontologist is studying an imprint of a raindrop left on a mud flat from a Jurassic thunderstorm. As she pauses for a drink of water from her canteen, she idly wonders how many molecules of that raindrop are in her canteen. Estimate this number, stating all of your assumptions.”
One student used the assumptions that there were 10,000 molecules in a raindrop, 1 million raindrops in a thunderstorm, and 1,000 years since the Jurassic, and that therefore there were about 100,000 molecules from the raindrop in the canteen.
10,000 in a raindrop? A raindrop is around 1 gram, water has a molecular number of 18. Therefore, a raindrop should have about 3 * 10^22 molecules… order of magintude my ass.
Also, there could be anamolies that throw your answer off. Frex, Bush Gardens in Florida and the San Diego Wild Animal Park in California both have several giraffes. Maybe one of them has an entire herd. Could bump the nuber of giraffes up to 50. For the record, my parents in north Georgia once photographed a kangaroo hanging out with a herd of deer in their front yard. It had escaped from a wild game reserve nearby. How many cases of stuff like that goes on unnoticed?
I think the point was to show an example of bad estimation. Every single number is horribly off.
(but at least the student did the smart thing and kept everything in powers of 10!)
Not to mention that a) the number of raindrops in a thunderstorm is completely irrelevant to the answer, and b) even if we accept the student’s numbers, there can’t possibly be 100,000 molecules from the raindrop in the canteen, if there were only 10,000 molecules in the original raindrop.
Not to mention:
[ul]
[li]Giraffutopia, the enormous giraffe sanctuary just outside Topeka, largest in the world.[/li][li]The pet giraffe population of suburban Albany, where quirky local tastes favor them over dogs and cats.[/li][li]Spotsylvania, Virginia — now inhabited almost entirely by giraffes, after mysterious beige pods began arriving on trucks a few months ago.[/li][/ul]
Because of all these anomalies, most people’s guesses of “a few hundred” giraffes here in the U.S. are woefully naive. Failing grades to all of you.
LOOKING NOTHING UP, NO REFERENCE MATERIALS:
A mole of water is 6.0210^23 molecules, and has a weight of ~18 grams (2 hydrogens at 1 each, 1 oxygen at 16). A gram of water is 1 cubic centimeter, so a raindrop is smaller than that, I’ll estimate at about 1/10 of a gram, that means 1/180 or 0.005 moles of water in a raindrop. That means 310^21 molecules of water in a raindrop.
Now, let’s assume that the total amount of water on earth since the Jurassic is constant, although of course the water molecules themselves have undergone some turnover. How much turnover? Hard to say, water is destroyed and created all the time in biological reactions. Hard to say exactly how much, but let’s ignore this, since at least half of earth’s water is probably fossil water. I think we can also assume 100% mixing in the 100 million years since the Jurassic.
So how much water is on earth? The earth is a sphere, 70% covered with water, estimate average depth of .5 km. When we know the surface area of the sphere we know the volume of water. I think the earth is 8000 km in diameter…ah, what’s the formula for surface area of a sphere? Can’t remember it, 4/3pi something. OK, suppose the earth is a cube with 8000 km sides. 6 sides, each with 6.410^6 km^2 area for total surface area of cube earth of 3.810^7 km^2. Sphere’s gotta be smaller than that, plus only 70% is water, so estimate 210^7 km^2 of water surface, with average depth of .5 km gives us 110^7 km^3 water volume.
How many moles is that? How many ccs in a km^3? Easy to answer, there are 10^6 cm^3 in a m^3, 10^9 m^3 in a km^3, so 10^12 cm^3 in a km^3. Each cc is ~.05 moles of water. So (10^7 km^3)(10^12 cm^3/km^3) gives us 10^19 grams of water on earth, or 510^17 moles. (510^17 moles)(6.0210^23 molecules/mole) gives us 310^41 molecules of water on earth.
So 310^21 molecules were in the raindrop, 310^41 molecules on earth, so a random molecule on earth has a 1 in 10^20 chance of being in that raindrop.
So, how big is the canteen? Let’s just say it’s a 1 liter canteen, which is 1000 grams of water, or 10,000 times bigger than a raindrop, or 3.310^25 molecules of water. 310^25 molecules, each with a 1/10^20 chance of being in the raindrop gives us 3^10^5 molecules
Therefore, there are probably on the order of 300,000 molecules from that Jurassic raindrop in her canteen.
A point you’re overlooking, or maybe I overestimate its importance, is this: how many water molecules that were around in the Jurassic are still in existence, with the same two hydrogen atoms stuck to the same oxygen atom? Water is pretty stable, but 150 million years is a long time.
A problem I heard once - you’re in a sailboat, sailing over the deepest part of the ocean, the Mariana Trench. A cannonball rolls off the deck at that point and plunges into the water. How long until it hits the bottom?
Did giraffe to mention that?
So… the point your making is that her guess actually was within an order of magnitude of your? :eek:
20 minutes in 1960?
Once.