Of course, where the heck am I supposed to get a math problem? I decided to use a combination of Bing and Wikipedia (I used Bing to find the Wikipedia article of which is excerpted):
The classic Fermi problem, generally attributed to Fermi,[2] is "How many piano tuners are there in Chicago?" A typical solution to this problem involves multiplying a series of estimates that yield the correct answer if the estimates are correct. For example, we might make the following assumptions:
- There are approximately 9,000,000 people living in Chicago.
- On average, there are two persons in each household in Chicago.
- Roughly one household in twenty has a piano that is tuned regularly.
- Pianos that are tuned regularly are tuned on average about once per year.
- It takes a piano tuner about two hours to tune a piano, including travel time.
- Each piano tuner works eight hours in a day, five days in a week, and 50 weeks in a year.
Okay, so let's hammer down that G. Polya layout.
- Understanding the Problem!
- Well, what do we want to solve for?
- We want to know "How many piano tuners there are in Chicago."
- So we know that our answer should be in the form of:
- "There are (blank) number of piano tuners in Chicago."
- By piano tuners, we're talking about the human beings that walk around tuning pianos!
- Well, what do we know?
- If we only take into account the question itself, I can't really do anything without making at least a few assumptions since I don't know anything about pianos, piano tuners, or Chicago other than the fact that Chicago is a big city somewhere in the northern region of the United States!
- The line after the question confirms this idea by claiming that a "typical solution... involves multiplying a series of estimates."
- So what are the estimates? Let's use the ones given! Let's simplify them for convenience and take all approximations as absolutes.
- There are 9 million people in Chicago.
- There are 2 people in each household in Chicago.
- 1 out of 20 households has 1 piano.
- No household has more than 1 piano.
- Pianos are tuned 1 time a year.
- Piano tuners spend 2 hours tuning pianos.
- Piano tuners spend all their work time tuning pianos.
- Each piano tuner works 8 hours a day.
- Each piano tuner works 5 days a week.
- Each piano tuners works 50 weeks a year.
- Okay, from what I can tell, that's everything we need to know!
- Devising a Plan!
- Find a connection between the data and the unknown.
- Okay, we know that we have a bunch of numbers for each estimate, and that each estimate seems to be related to the one before it!
- For example, there are 9 million people in Chicago.
- There are also 2 people in each household in Chicago.
- We can assume that Chicago is just stating our domain, so we know we're dealing with the same people in the same city!
- Okay, so if there are are 2 people per household, then our number of households must be half the number of people!
- So we know that there are 4.5 million people in Chicago!
- So we can maybe solve for each newly introduced variable (like the number of households, the number of regularly tuned pianos, etc.) with information in the previous line!
- Plan A!
- We can convert each bit of information into a ratio and use dimensional analysis to get an answer!
- The numbers may be big, so a calculator will help!
- Carrying Out the Plan!
- Okay, let's convert everything into easy to dimensional analyze ratios.
- (9 million people)
- (2 people)/(household)
- (1 piano)/(20 household)
- [(1 tune)/(year)]/(piano)
- (2 hour)/(tune)
- (8 hour)/(workday)
- (5 workday)/(workweek).
- [(50 workweek)/(year)]/(piano tuner)
- So there are a few too many parentheses and not enough numbers. Let's simplify them.
- 9,000,000 people
- 2 people/household
- 20 household/piano
- tune/(year*piano)
- 2 hour/tune
- 8 hour/workday
- 5 workday/workweek
- 50 workweek/(year*(piano tuner))
- And now let's multiply them all together remembering that each ratio's numerator is equal to the denominator.
- (9,000,000 people) * (household/(2 people)) = 4,500,000 household
- (4,500,000 household) * (piano/(20 household)) = 225,000 piano
- (225,000 piano) * (tune/(year*piano)) = (225,000 tune)/(year)
- (225,000 tune/year) * (2 hour/tune) = (450,000 hour/year)
- (450,000 hour/year) * (workday/(8 hour)) = 56,250 workday/year
- (56,250 workday/year) * (workweek/(5 workday)) = 11,250 workweek/year
- (11,250 workweek/year) * ((year*piano tuner)/(50 workweek)) = 225 piano tuner
- So our result is 225 piano tuners. Can we put it into our answer sentence?
- "There are 225 piano tuners in Chicago."
- Yup, this is the right answer!
- Looking Back!
- Can we check our result? The answer is yes. We can check it with another person by reading the Wikipedia article and seeing that they also got 225 piano tuners.
- The actual number is reported to be 290!
- From these estimates, we can see that even though we didn't know exactly what the answer is, we landed at a pretty close result just from making approximated assumptions.
- However, upon closer examination, our margin of error is approximately 23%. That's pretty high!
- Imagine being in charge of estimating the number of people that will vote for a certain person during the Presidential elections (in the United States). Do you really want to be that one guy that miscounted 23% of the polls? I heard margins of error for those polls tend to be around 5-15%!
- And do you want to be that person that miscounted for 23% of the people that desperately need supplies after a natural disaster? If there were hundreds or thousands of people, you could potentially kill hundreds or thousands of lives easily.
- So what does this tell us? I think this means that although Fermi was awesome at estimating for a small number of people, estimations like this are bad if you're trying to estimate a larger population!
And I like how I can use this proof as a summary for this class!:
http://deerickchau.blogspot.ca/2014/10/oct-6-to-oct-10.html
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