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P9: Give or Take (Back-of-the-Envelope Estimates / Fermi Problems)
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How many piano tuners are there in New York City? How much cheese is there in Delaware? And how can you find out? All of this and more on this problem-episode of Breaking Math.
This episode distributed under a Creative Commons Attribution-ShareAlike-Noncommercial 4.0 International License. For more information, visit creativecommons.org
Featuring theme song and outro by Elliot Smith of Albuquerque.
[Featuring: Sofía Baca, Meryl Flaherty]
Recommended
Transcript
Introduction to Fermi Problems
00:00:00
Speaker
How many piano tuners are there in New York City? How much cheese is there in Delaware? And how can you find out? All this and more on this problem episode of Breaking Math. P9, give or take.
00:00:19
Speaker
I'm Sophia. I'm Meryl. And you're listening to Breaking Math. First, just a few plugs. The Tensor poster, 24 by 36 inches mat about tensors which are used in general relativity. You can buy that at our Facebook poster store at facebook.com slash Breaking Math podcast. If you want to donate monthly, which we really appreciate, you'll get our episodes without ads and even have the outlines that we use for the show.
00:00:41
Speaker
One dollar or more gets you that and that's at patreon.com breaking math and we really appreciate that Oh the tensor poster. It's also there as a prize for donating Pi to the e dollars I believe if you want to buy it directly from Facebook though It's fifteen dollars and fifteen cents plus four dollars and fifty cents shipping and handling for a total of nineteen dollars and sixty-five cents You can also find updates about the show there
00:01:03
Speaker
and on twitter at breakingmouthpod, our website breakingmouthpodcast.app, and you can email us at breakingmouthpodcast.gmail.com with questions, comments, ideas, and corrections.
Origin and Purpose of Fermi Problems
00:01:15
Speaker
Anyway, so for many problems, what are they? So they're a way of using dimensional analysis to sort of get, you know, an estimate of, you know, certain problems like how many piano tuners are there in New York.
00:01:29
Speaker
Yeah, and these are problems that were made popular by Fermi during World War II. They would actually have competitions where people would try to get something correct to within one digit within 10 seconds and all kinds of silly things like that. But Fermi, when the bomb went off, he dropped pieces of paper from his hands and he measured how far they fell from the top of his head.
00:01:52
Speaker
and using that he got that the bomb at the Trinity site was 10 kilotons and its actual value is 21 kilotons but that's the kind of accuracy we're talking about here right like within an order of magnitude exactly and another name for these by the way is back of the envelope estimates of course just because they could be done on random scraps of paper napkin estimates i believe is also a term there's all sorts of terms but yeah so how do we do the piano tuners there on the New York City thing
Estimating Piano Tuners in New York City
00:02:21
Speaker
Well, first we're going to start with, so how many people are there in New York City? There's like, there's, there's, I mean, it depends on what you're talking about, the regular city or the metropolitan area, but we're going to say 10 million people roughly. Okay. It's definitely in that order of magnitude. Yeah. Cause it's not a million people cause it's not some burg and it's not a hundred million people because the cities like that are still fictional.
00:02:44
Speaker
So how many pianos per person do you think there are? Because most pianos are probably in houses, given the fact that concerts and stuff like that have a piano that could be used over and over again. I'd say maybe one piano to every hundred people, because I've only been in a couple of houses that have pianos in them. Okay, so that means that with 10 million people, there's about 100,000 people in New York City who have pianos.
00:03:12
Speaker
Yeah, so there's 100,000 people who have pianos, which is like 100,000 pianos that need tuning. So now we're getting somewhere, right? We know how many pianos roughly need tuning. So how frequently are they going to be tuned? Probably once every year to five years. Yeah, so it's around two and a half years. And what's 100,000 pianos divided by two and a half years per tuning? So that would be 40,000 tunings a year.
00:03:41
Speaker
Yeah, 40,000 tunings a year because you need one tuning per piano. Let's say somebody needs to make a living tuning, right? Yeah. They probably need to work five days a week for roughly 50 weeks or do about, let's say 250 tunings per year, roughly. Because I know that it takes a while to tune a piano. So what's 40,000 divided by 250? So that's about 160 is what we could guess is how many piano tuners are in New York City.
00:04:09
Speaker
Yeah, and of course that's either a little high or a little low. And the thing is, you might have noticed that we'd made a bunch of steps, but for a reason that we'll tell you later, the amount of steps that we take does not actually make our guess that much worse because of something called random walks and because of the concept of logarithms.
00:04:28
Speaker
But we'll talk about that later. I took on to myself to look up how many piano tuners there were in New York City. And I got a listing for about 20 businesses. A lot of them were single people who did it. And a few of them were places that were known as like, you know, tuning quote unquote institutions.
00:04:46
Speaker
So it looks like there's about 50 piano tuners in New York City, which means they're not that far off. It also turns out that if you start calling piano tuners in New York as soon as they open up, you might get hung up on a couple of times. This is how I spent my morning. And we haven't prepared any more back of the envelope estimations for the show, but we're gonna do some on the show just because they're fun to do and to show you how impromptu these can be done. So, um, I don't know, let's think of another one. Do you want to think of this next one, Meryl?
Cat Ownership and Litter Boxes in Austin
00:05:14
Speaker
Um, let's think about how many litter boxes are there in, let's say Austin, Texas. All right. Austin, Texas. I know that's a biggish city, but not that big. So let's say 2 million people live there. Yeah. Closer to, closer to 1 million, I would say. Yeah. So let's start with 1 million since if we have information, let's use it obviously. So let's say 1 million people. I'd say roughly 1 in 10 people probably own a cat. Sounds fair enough.
00:05:43
Speaker
All right, so you have a million people divided by 10 per people per ollie per every cat. That's probably a hundred thousand cats. And there's probably one litter box per cat. So somewhere around a hundred thousand litter boxes in Austin. Right. And that's saying that's assuming we have one litter box for a household with at least one cat. Yeah. And I know that's recommend that you have as many litter boxes as you have cats plus one, but I know not everybody follows that.
00:06:11
Speaker
All right, awesome. Let's think of another one quick. So here's one.
Electricity Usage in Moscow
00:06:16
Speaker
How much electricity and we'll say in kilowatt hours is used in the city of Moscow?
00:06:23
Speaker
Right, so let's see. You have electricity used by houses and you have a certain amount of electricity used per person in houses, and we have electricity from industry, right? So let's figure out the houses first. Let me think. Okay, so there's probably about 5 million people in Moscow or something like that. I'm guessing. I'm going to say 5 million people. Okay, let's just go with that guess.
00:06:48
Speaker
All right, so 5 million people in Moscow. Let's see. All right, so let's say that the amount of electricity that you use in a house is the same as, I don't know, 1060 watt bulbs on 24 hours a day per person. That sounds reasonable.
00:07:12
Speaker
Yeah, we'll roll with it. All right, so let's see. And let's see, let's say industrially, just knowing how much industry takes up of electricity and stuff, let's just multiply that by five. So for each person, let's say that they consume an average of 50, 60 watt bulbs at any given time, which is the same thing as
00:07:40
Speaker
3 a 3 kilowatt bulb being on 24 hours a day 3 kilowatt so that times 24 that is alright, so we have 70 kilowatt hours per person per day times 5 million people and a million times a thousand is a billion so we're using gigawatts, so that'd be about 350 gigawatt roughly
00:08:03
Speaker
And I decided to look up how many people... about the electricity consumption in Russia. And the electricity consumption of the entire country of Russia is about 145 gigawatts per day. And let's see how many people live in Moscow. And a little less than a tenth of the population lives there. So we're a little high, high by a factor of about 10, right? Yeah. But we weren't that far off, especially from the little tiny bit of data that we had.
00:08:32
Speaker
Yeah, and I just made that problem up and we didn't Google anything before. Yeah, and it shows really the value of this sort of reasoning. And we'll come back to a few more of these problems. But before we go on with these, we're going to justify the reason why we can do this. And you can either sit through the explanation or you can fast forward a little bit. Stay tuned.
00:08:57
Speaker
So to understand the reasoning behind Fermi estimates, we're just going to talk a little bit about addition, multiplication, and logarithms.
Understanding Logarithms
00:09:05
Speaker
All right, so this could choose some intuition, dear audience. I'm a writer from the 19th century. So let's start with the number one. If I multiply that number by 10, I'll have 10, right? Yeah. Which has one zero. So we started with no zeros, and now we have one zero. So we can't do much with just that. But if I multiply by 1,000, which has three zeros, we'll get 10,000, which has four zeros, and four is three plus one.
00:09:34
Speaker
And if I multiplied that again by a number with five zeros, we would get a number with five plus four equals nine zeros, right? Yeah, and so there's actually a name for the number of zeros that, you know, a number has. And we call that, or at least the number of digits, and we call that the order of magnitude.
00:09:55
Speaker
Yeah, and that's what it means when we say that back of the envelope estimates are roughly order of magnitude estimates because it turns out that the number 10 is also the number of fingers that we have, but it's also around how useful an estimate can be if it's just a really broad estimate.
00:10:11
Speaker
And so you notice that obviously multiplication and addition seem to be related by adding zeros and subtracting zeros by dividing. And of course, what is this relation between addition and multiplication? How can it be typified? Let's give a concrete example. 4 is 2 squared, right? And 16 is 2 to the 4th, right? Yeah.
00:10:35
Speaker
And 4 times 16 is 64, which turns out to be 2 times 2 times 2 times 2 times 2 times 2, or 2 to the 6th. And 6 is 2 plus 4, right? Sounds right.
00:10:46
Speaker
Now, let's say we want to go backwards, right? We can use what we call logarithms. So logarithms answer the question for... Let's say we have two numbers, right? Let's say we have the number 10 and we have the number 1000, and we want to know to what power do we raise 10 to get 1000?
00:11:07
Speaker
We know the answer is three, but if we wanted to write this down mathematically, we would say log base 10, which is just LOG with little 10 next to the G, little small 10 in the subscript. And then next to that, you just write 1000. So that says log 10 of 1000, and then that equals three, right? Right. So what would be log base two of 1000?
00:11:31
Speaker
So let's start with, let's take two to the ninth, that's 512. Two to the 10th then is 1024. So what we get then is that, so two to the 10th is really close to 1000. That means that log two of 1000 is probably really close to 10.
00:11:50
Speaker
Yeah, and the actual answer is 9.96. And it's actually sometimes used in back of the envelope estimations. We won't show exactly that could be maybe a take-home problem. But using the fact that log base two of 10 is roughly three and a third, you could do a lot of cool stuff with that.
00:12:10
Speaker
And so let's put this all together. Logarithms also have a relationship between addition and multiplication. And that relationship is log base a of b times c is equal to log base a of b plus log base a of c. So when you multiply the numbers that you're taking the arguments of, all you do is add the numbers together.
00:12:32
Speaker
And that's how slide rules work, right? You just keep multiplying numbers and like if I keep multiplying by two on a slide rule, I keep going down the slide rule the same distance, but the numbers will keep doubling and doubling because the numbers keep getting smaller and smaller, right? Right.
00:12:49
Speaker
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Speaker
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00:14:23
Speaker
Now we're going to be talking a little bit about random walks.
Random Walks and Expected Values
00:14:27
Speaker
So simply put, a random walk is like, let's say you start at a certain position and you flip a coin. If the coin flips to heads, you walk a foot forward. If it flips to tails, you walk one foot backward.
00:14:43
Speaker
Now, if you keep doing that over and over again, you're going to be describing a random walk. And random walks describe everything from stock markets to Brownian motion, which is like if you put pollen particles on top of water so that they float on top, and you look at them under a microscope, you'll see that they diffuse out without any extra energy, and that is the Brownian motion.
00:15:08
Speaker
But to talk about random walks, we're going to have to talk about the components of random walks, which are called random variables. So random variables are simply put variables with a range of outcomes. What are some examples of this? Obviously coin flip and die rolls, dart throws, coordinates, what else? The weight of a random person. If you spin a spinner for a board game, what color to land on?
00:15:30
Speaker
Yeah, that's another random variable. Yeah, there's all sorts of random variables. And random variables happen when you have a lack of information about something, which is kind of appropriate for the episode that we're doing, right? Right. Because the whole point of random variables is that you don't know what the result is going to be until it happens.
00:15:48
Speaker
Yeah, and the whole point as well as Fermi estimations is that we don't know all this stuff. I mean, we know a certain amount, but quantifying on which ignorance we have is important, especially to convince ourselves that the error doesn't just stack up, right? Because let's say that we're off by a factor of two each time and we just kept getting worse and worse estimations, then within five steps, we'd be already at 32 times less than or more than the value. We would just keep growing exponentially, basically.
00:16:16
Speaker
Random variables have a few properties. One of those is the expected value. So that's kind of like the average outcome of a random variable. So the expected value of a die would be what, Meryl? So we want to average together all the numbers on a die. So to do that, we add them up. So 1 plus 2 plus 3 plus 4 plus 5 plus 6. And that adds up to 6 times 3 plus 3, which is 21.
00:16:43
Speaker
Right. And then we want to divide that by six. Yeah. So we get 21 sixths or seven halves. So three and a half.
00:16:55
Speaker
Yeah, which makes sense because the lowest one is one and the highest one is six, so of course it's going to be somewhere in the middle there. What's interesting too is that if you flip two dice, the most common number that you're going to land on is seven, which is also the average outcome as well, the expected value as well in that situation.
00:17:17
Speaker
And it turns out that if you add random variables, you add their expected values together. Now we're going to talk about another metric associated with random variables called standard deviation. Standard deviation is a measure of how widely spread a random variable is. If a random variable has a standard deviation of zero, then it's completely deterministic. And it is the average, basically, the square root of the average of how far each outcome is from the average outcome.
00:17:43
Speaker
But before we talk about that, we're going to talk about bell curves, right? Do you want us to talk about what a bell curve is really quick? So if we have a lot of random variables that we're putting together, then a lot of times our probabilities become this distribution that's called a bell curve or a normal distribution.
00:18:00
Speaker
Yeah and that's found in a lot of kind of like evenly distributed data and one thing is a height. So the European female mean height was around 5 foot 5.4 inches or 164.7 centimeters with a standard deviation of 2.78 inches or 7.07 centimeters. So what can we do with this information?
00:18:22
Speaker
So we could find out however many European women are within a certain range of height. Yeah. And there's formulas that you could plug into this like the error function and stuff like that to get this. But one handy rule is the 6,895,99.7 rule.
00:18:39
Speaker
And so what that means, so 68% of a population or a number of samples are going to be within one standard deviation of our average 95% within two standard deviations and 99.7 within three.
00:18:58
Speaker
Yeah, and so three standard deviations in this situation would be about 21.21 centimeters. And so if we add and subtract that from the mean height of 164.7 centimeters, we get a height anywhere between 157 and 171 centimeters or 4 foot 9 inches and 6 foot 1.8 inches. And that range contains 99.7% of European women.
00:19:23
Speaker
And you could also divide that by 2, so you could say that 99.85% of women are less than 6 foot 1.8 inches tall, which is kind of interesting. So how do we calculate the standard deviation?
00:19:38
Speaker
The standard deviation is the square root of the average of how far the outcomes are from the average outcome, squared. Yeah, and I know that sometimes you divide by, instead of, you know how like in an average you divide by n. I know that sometimes with the standard deviation you divide by n minus one, but we're not going to be messing with that. That's called the adjusted.
00:19:58
Speaker
standard deviation and it happens because of degrees of freedom. Basically when you average out a certain number of samples you're always going to get a zero term so you're going to calculate the standard deviation to be a little bit too low. But we're going to ignore that for the time being. It's just something good to know so that you don't take this definition as the absolute one in all circumstances. So what's the standard deviation from average die roll?
00:20:21
Speaker
So we know that every outcome is going to be some distance from seven halves or three and a half. Yeah, which is what our expected value from earlier. And so what we're going to do is, so it's going to be one sixth times each of those outcomes minus seven halves squared, and then all under a square root.
00:20:45
Speaker
Yeah, which comes out to the square root of 35 twelfths or roughly 1.7. And the standard deviation of a random variable that's uniformly distributed between negative one and one is a square root of one third. And a uniform distribution just means kind of like the wheel of fortune or like something like that where you have a whole range of something between like zero and one, right? Zero and one turn of the wheel of fortune. Yeah, so every outcome is equally likely.
00:21:11
Speaker
And of course, you could do arithmetic with random variables. So if you multiply two random variables with their certain variances, you multiply their variances, which also means you multiply their standard deviations,
Logarithmic Operations as Random Walks
00:21:20
Speaker
right? Right. And the expected value of this new random variable is the product of those expected values too. It works somewhat with addition, but you also have to add in twice the covariance, so we're not going to be messing with that. And with Fermi estimates, we don't usually throw in addition unless we're doing it as part of
00:21:39
Speaker
Something bigger and you were usually pretty sure about something if we throw in addition, right? like I mean remember the Moscow problem we had to throw in addition for the For industry, but that was only it was only really multiplying something by a guess, right? Right. I
00:21:53
Speaker
So a random walk is the sum of an output of a bunch of random variables. So it's like we said earlier, flipping a coin repeatedly and going forward one foot if it's heads and going backward if it's tails, or Brownian motion. Because Brownian motion is the sum of all the momentum changes that you get from particles, like gas particles, or like water particles hitting an object. That's why tiny objects are affected by Brownian motion, but the bigger the object it's less affected.
00:22:23
Speaker
So you might be wondering how far will we get with a random walk? So it turns out, and this is something that dates back to Einstein's paper on pollen on the surface of water, the stuff starts spreading out immediately but then it slows down its spread as everything goes randomly and it turns out that the standard deviation basically increases with the square root of time with Brownian motion or with the square root of how many trials, you know?
00:22:52
Speaker
Yeah, so what I'm getting at is that standard deviation is a bit more resistant to adding different random steps together.
00:23:02
Speaker
Yeah, it's almost like they correct each other but it's weird and it's also kind of weird too because the expected value of the sum of the trials is zero because it's equally likely to be on either side but the expected value of the square of the sum is n. We might do a whole episode on that but the takeaway is that with a random walk it'll be harder and harder to expect further and further distances over time in proportion to the square root of n.
00:23:28
Speaker
After one second, you might expect a certain proportion within a certain distance. After four seconds, you increase the distance by one unit of distance. After five more seconds, which is nine, you just go one more unit and so forth. So we now know that random walks can only get so far over so much time. And for me, approximations have a certain standard deviation as well. And we know about logarithms. So how can we put those all together?
00:23:56
Speaker
So, at the end of the day, a Fermi approximation is a random walk. Yeah, and you might wonder, like, how is that? Well, it's mostly multiplying and dividing each time, and we're all over and underestimating. Basically, what we're doing is we're adding the logarithm of what we're adding and dividing. Because remember, on a slide rule, multiplying and dividing are just moving left and right by the amount that you're multiplying and dividing by on the rule.
00:24:19
Speaker
And so it's basically a random walk right on a slide rule. And so that means that after n multiplications, let's say we're off by a factor of at most two each time. So we're either half under or half over. Then we expect with a standard deviation that corresponds to square root of n times the average standard deviation. And remember our standard deviation here is us getting a value between one half and two. So that means after nine steps, what's the square root of nine? Three.
00:24:48
Speaker
three and since we're off by a factor of two each time we raise two to the power of three or eight. So we're within roughly one eight to eight times the true value with the same amount of confidence that we were within one half to two times the value and that's after nine whole steps. After 20 steps our approximation is good to basically two orders of magnitude or basically 120 seconds which is two digits you know.
00:25:14
Speaker
So even if we're less accurate then, so say we're within a factor of three, so those numbers would be 27 and 136. Yes, it's so weird. That's the amazing thing about this is that even though for the most part, we're totally guessing numbers, well, we're taking an educated guess. But at the end of the day, we get something that looks like a reasonable estimate.
00:25:38
Speaker
Yeah, which is kind of insane, but now be the annoying person at parties who goes around asking, Hey, how many of ice cubes are there in
Estimating Ice Cubes in Memphis
00:25:45
Speaker
Memphis? How many ice cubes are there in Memphis? Let's see. So how big of a city is Memphis? I think it's like a bit, I mean, rock and roll comes from there. So I assume it's like a couple million at least. Yeah. Let's go with a million. All right. So it's about a million. Um, how many ice cubes do you think a typical person will use in a day?
00:26:05
Speaker
I'd say let's say let's say 10 ice cubes because some people really like their sodas. Okay, so a million people times 10 ice cubes is the city goes through 10 million ice cubes a day. So if the city goes through 10 million ice cubes a day, then we just need to take a guess of how many more times of that do we need in order to satisfy the need and keep enough stored. So let's guess that there's 10 times as much of that. So we'll say that there's 100 million ice cubes in the city of Memphis.
00:26:35
Speaker
Yep, and what is the weight of those ice cubes? I'll do this one. There are a few cubic centimeters of water, so let's say 10 grams. 10 grams times 100 million is a billion grams, which is a million kilograms, which is of ice or about a thousand tons of ice. All right, let's see.
Theoretical Bug Spray Needs in the Amazon
00:27:00
Speaker
How much bug spray would it take to kill all the bugs in the Amazon?
00:27:04
Speaker
Ooh, that's a good one. So how big do we want to say the Amazon is? Let's say Brazil is about the size of the Louisiana Purchase, which I know is a million square kilometers, give or take. So let's say the Amazon is what, like a 10th of Brazil's landmass? Something like that. So a hundred thousand square kilometers?
00:27:27
Speaker
Yeah, let's say that, like 100,000 square kilometers, give or take, in the Amazon. Now, how many bugs, obviously, per square kilometer? And I guess we could think about that in terms of square meters. So let's do maybe bugs per square meter. We know that the Amazon is a rainforest, which means that it's not particularly rich in minerals, which means that it's not going to host a gigantic insect population but a respectable one. So I'd say one per square meter is something, give or take, wildly.
00:27:56
Speaker
Yeah, I would have guessed maybe 10. Okay, so let's take the average, the log average and do three. Okay, so three bugs per square meter.
00:28:05
Speaker
So if it's three bucks per square meter, then that is three million per square kilometer. So three million per square kilometer times a hundred thousand square kilometers is, let's see, that's three times 10 to the sixth times one times 10 to the fifth, which is three times 10 to the 11th or 300 billion bugs.
00:28:30
Speaker
All right, that's part one. So there's 300 billion bugs. So how much bug spray would it take to kill 300 billion bugs? So I guess the way to think about it is let's say that you just see one bug and you need to kill it. So let's just say that you have one bug and you want to spray it dead. So how much bug spray like do you think comes out of the can whenever you spray one bug?
00:28:50
Speaker
Um, if I take a drop of water, which is about a 10th of a CC, I just happened to know that and I spread it around. It's probably, probably about the same area as a bug spray. Same volume. You mean? Oh yeah, sorry. Same volume as a spray of bug spray. Same kind of wetness, you know? Okay. So we're saying like 10th of a milliliter for a single blog. Yeah. So we have a 10th of a milliliter times what?
00:29:14
Speaker
So a tenth of a milliliter times how many how many bugs did we say 300 billion 300 billion or million a billion because 300 because there's 100,000 square kilometers times 3 million bugs per square kilometer.
00:29:28
Speaker
Okay, so it would take then, so 30 billion milliliters or 30 million liters of bug spray to kill off the entire bug population of the Amazon. And as a note to the listener, we do not condone killing off the entire insect population of the Amazon. Yeah, save that for Mars. I don't know. But seriously, don't do that. Try to be environmentally conscious while writing letters to those who have more power than you and also direct action.
Conclusion and Reflections on Fermi Estimates
00:29:58
Speaker
So yeah, that's, for me, approximations. What did you think of this? I know that you were slightly newer to these than I was. Yeah, I am not a statistician, and in general, I'm not the biggest fan of stats, but I thought that this was at least fun.
00:30:12
Speaker
Yeah, and it's fun being able to answer dumb questions like how many rubber bands are there in Idaho, which could be a take-home quiz. Another take-home quiz could be how many batteries have ever been consumed since the modern batteries have been around? How much money was spent in the 1930s in the United States? And let's see, what's a good one?
00:30:37
Speaker
Or maybe how many calories does America eat together on Thanksgiving? Yeah, so I am Sophia. I'm Meryl. And this has been a breaking math problem episode. Patreon would be great. We have a couple of new patrons and thanks to them so much. And we hope to get more because we want to be able to do some cool stuff for you and that provides a reliable source of income. So thank you for everyone who is a patron.
00:31:03
Speaker
I will give a shout out to the University of New Mexico Grad Workers Union. UNM works because we do.