GuesstimatesSome solutions for the book “Guesstimation” by Weinstein & AdamAnton Antonovantononcube@gmail.comhttp://www.linkedin.com/pub/anton-antonov/7/879/89a

IntroductionIn this notebook (article) I have shown the solutions of some problems from the book “Guesstimation: Solvingthe World’s Problems on the Back of a Cocktail Napkin” by Lawrence Weinstein and John A. Adam. ( Seehttp://www.amazon.com/Guesstimation-Solving-Worlds-Problems-Cocktail/dp/0691129495 .)

I have followed the chapter structure of the book. I have copied the questions from the corresponding sections.I give my estimates and the ones produced and explained in the book.

The most interesting part of this notebook (article) is the example and explanations I give of why using thegeometric mean to estimate averages from lower and upper bounds estimates is adequate in many cases. Seethe sub-section “Geometric mean” in the next section. In the first chapter of the book the authors just explain atechnique for quick computation of the geometric mean. They do not elaborate much on why it is a betterestimate except by giving few examples. (I found their short description inspiring, so I looked for an explana-tion and other examples.)

1. How to Solve Problems

ü Geometric mean

I show in this sub-section an example of why it is better to use the geometric mean to calculate averages basedon estimated minimum and maximum values.

Suppose we have to estimate the mpg (miles per gallon) of the average car in US. (See problem “5.2 Drowningin gasoline“ below.) We can use the estimates 6 and 50 as the maximum and minimum mpg’s respectively.(The number 6 is for really old or antique cars; the number 50 is for Toyota Prius and the like.) Using thegeometric mean we get 6 µ 50 º 17.3 to be the estimate of the average US car mpg.

Now let us try to make a more informed estimate using mpg’s for variety of car models. We will assume thatall models are used equally likely.

We have mpg data for 32 car models.

I have taken the data from R with the commandsapply(list(row.names(mtcars),mtcars$mpg),unlist) .

Some descriptive statistics of the mpg data:

Min Max Mean Median StandardDeviation10.4 33.9 20.0906 19.2 6.02695

This histogram shows the probability to pick a car with a given mpg:

Obviously the data fits better an exponential law than a linear one. In the graph below, the red line is for anapproximation with a function of the type a+ b 1.06x. The blue line is for the type a+ b x.

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0 5 10 15 20 25 30

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30

Let us calculate the geometric and arithmetic means based on the minimum and maximum values.

H* geometric mean *L

gm = 10.4 * 33.9

18.7766

H* arithmetic mean *L

am =10.4 + 33.9

2

22.15

If we place this numbers in the histogram for the mpg distribution it can be easily seen that the arithmeticmean overestimates the average mpg and the geometric mean is a quite good estimate. (Red line for geometricmean, blue line for arithmetic mean.)

Let us look at the ratios between the sorted in increasing order mpg’s:

Min Max Mean Median StandardDeviation1. 1.27885 1.04003 1.02674 0.0527335

We can say that with the geometric mean we have approximated the following formula:

H* minimum mpg µ ratios mean ^ half of the total number of mpg increments *L10.4 * Mean@ratiosD^15

18.738

Another way to look at the geometric mean is that it approximates the median value just by using the mini-mum and maximum values. As an estimator for incomplete and missing data, the median is more robust thanthe mean. In the plot below it can be seen (red grid lines) that the geometric mean 18.7766 is closest to thevalue of the median of the mpg data.

Guesstimates.nb 3

Another way to look at the geometric mean is that it approximates the median value just by using the mini-mum and maximum values. As an estimator for incomplete and missing data, the median is more robust thanthe mean. In the plot below it can be seen (red grid lines) that the geometric mean 18.7766 is closest to thevalue of the median of the mpg data.

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I would conjecture that in most cases when we bring minimum and maximum values to estimate a value we doso because we have not build a good intuition and judgment for value’s spread and distribution. If the valueobeyed a linear law we would have noticed that and would be inclined to use the arithmetic mean with highercertainty. In all other cases we are better off using the geometric mean. This is because, say, life phenomenastend to follow exponential laws, and/or many distributions have exponents in them.

To support the last statement in the previous paragraph look at:-- Pareto principle -- http://en.wikipedia.org/wiki/Pareto_principle ;-- Long tail -- http://en.wikipedia.org/wiki/The_Long_Tail ;-- Power law -- http://en.wikipedia.org/wiki/Power_law .

ü Height of stack of 10^8 tickets

Question: Your chance of winning the MongaMillions lottery is one in 100 million. If you stacked up all thepossible different lottery tickets, how tall would this stack be? Which distance is this closest to: a tall building(100 m or 300 ft), a small mountain (1000 m), Mt Everest (10,000 m), the height of the atmosphere (10^5 m),the distance from New York to Chicago (10^6 m), the diameter of the Earth (10^7 m), or the distance to themoon (4×10^8 m)?

H*thickness of 10 tickets µ 100 million *LH0.001 m ê 10L * 10^8

10 000. m

So it is close to Mount Everest.

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2. Dealing with Large Numbers

3. General Questions

ü 3.1 One big family

Question: If all the humans in the world were crammed together, how much area would we require? Comparethis to the area of a large city, a state or small country, the US, Asia.How much area would we need if we gave every family a house and a yard (i.e., a small plot of land)?

H* world population µ human hight µ width µ side *L6.93 * 10^9 * H1.6 * .7 * .5L m^3

3.8808 µ 109 m3

H* world population µ human width µ side *L6.93 * 10^9 * H.7 * .45L m^2

2.18295 µ 109 m2

H* world population µ area to live *L6.93 * 10^9 * H2 * 2L m^2

2.772 µ 1010 m2

ü 3.2 Fore!

Question: How many golf balls would it take to circle the Earth at the equator?

Golf ball diameter:

0.02 * 0.04

0.0282843

Earth diameter:

ed = 100 000 * 10^3 * 400 000 * 10^3

200 000 000

2 * p * ed ê 2.0

6.28319 µ 108

% ê 0.03

2.0944 µ 1010

My first un-informed answer is 2µ 1010 golf balls.

Earth equatorial radius from WolframAlpha.com is 6378.14 km.

Guesstimates.nb 5

er = 6378.14 µ 10^3

6.37814 µ 106

Several estimates to answer the question:

H* Earth's equator lengthêgolf ball diameter *L2 * p * er ê 0.03

1.33583 µ 109

2 * p * er ê 0.04

1.00188 µ 109

2 * p * er ê 0.02

2.00375 µ 109

So my answer is 1.5µ 109 golf balls.

The answer in the book is 109 golf balls.

ü 3.3 This is a fine pickle youʼve got us into, Patty

Question: If all the pickles sold in the US last year were placed end-to-end, what distance would they cover?

300 000 000 * 2 * 200 * 0.03 * 0.12

3.6 µ 108

My answer º1.2 light seconds.

The answer in the book is 6µ 108 m or º 2 light seconds.

ü 3.4 Throwing in the towel

Question: What is the surface area of a typical bath towel (include the fibers!). Compare this to the area of aroom, a house, a football field.

ü 3.5 A mole of cats

Question: How massive is a mole of cats? (A mole is the number of atoms that weigh that element’s atomicweight in grams. For example, a mole of hydrogen weighs 1 gram and a mole of carbon weighs 12 grams. It isused in chemistry to make sure that there are equivalent numbers of atoms for a chemical reaction.) Comparethis to the mass of a mountain, a continent, the moon (7 × 1022 kg), the Earth (6 × 1024 kg).

H* Avogadro's constant µ cat mass estimate *L

6.022142 * 10^23 * 0.2 * 14.00 kg

1.0077 µ 1024 kg

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H* Avogadro's constant µ 4 kg *L6.022142 * 10^23 * 4 kg

2.40886 µ 1024 kg

H* my first cat mass estimate *L

0.2 * 14.00 kg

1.67332 kg

The answer in the book is 5µ 1024 kg.

ü 3.8 Tons of trash

How much domestic trash is collected each year in the US (in m^3 or tons)?

H* US population µ 365 µ trash per person in kg *L

300 µ 10^6 * 365 * 1.0 * 12.0

3.79319 µ 1011

H* the result above in tons *L% ê 1000

3.79319 µ 108

My answer is 4µ 108 tons.

300 µ 10^6 * 365 * 0.01 * 0.05

2.44849 µ 109

My answer is 2×109 m3.

The answers in the book are:-- 3µ 108 m3 of trash/year;-- “According to the US Environmental Protection Agency [9], in 2005 the US generated 245 million(2.45µ 108) tons of municipal solid waste (including recycling).”

4. Animals and people

ü 4.1 More numerous than the stars in the sky

Question: How many cells are there in the human body?

H* cell size *Lcs = 10. ê 10^6

0.00001

Guesstimates.nb 7

H* average human body volume ê cell volume *L0.0664 ê Hcs^3L

6.64 µ 1013

H* my body volume ê cell volume *L0.09 ê Hcs^3L

9. µ 1013

The answer in the book is 1014.

ü 4.2 Laboring in vein

Question: What is the total volume of human blood in the world?

ü 4.6 Playing the field

Question: How far does a soccer or field hockey player travel during the course of a 90-minute game?

H*number of sprints per game µ length of a sprint, m *L

ns = 20 * 50 14 * 60 êê N

916.515

The distance above is too little.

H* meters run in 10 seconds *L

14 * 60 êê N

28.9828

H* game time in minutes µ 6 µ 10 second run distance *L90.0 * 6 * %

15 650.7

My answer º 16 km. Seems right.

The answer in the book is 20 km.

ü 4.7 Eww... gross

Question: How many people in the world are picking their nose right now?

H*average time of picking nose in seconds *L

2 * 20.

6.32456

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H*number of nose picking times per day*L

1 * 6 êê N

2.44949

H* world population *L6.84 * 10^9

H* population µ probability a person to pick his nose during the day *L

6.84 * 10^9 *H6 * 2.5L

15 * 3600

1.9 µ 106

My estimate of the number of people picking their noses at every second is 2×106.

The answer in the book is 107.

ü 4.9 Letʼs get one thing straight!

Question: How long is all the DNA in your body? How long is the DNA of all humanity?

5. Transportation

ü 5.1 Driving past Saturn

Question: How many total miles (or kilometers) do all Americans drive in one year? How does this compareto the circumference of the Earth (2.5 × 10^4 mi), the distance to the Moon (2.4 × 10^5 mi), the distance to theSun (9 × 10^7 mi), or the distance to Saturn (10^9 mi)?

H* number of employed people in USA *L130 000 000

H* number of people µ commute distanceestimate in km µ number of days per year *L

130 000 000 * J 2 * 120 * 1.6N * 365

1.17615 µ 1012

H* the result above in miles *L% ê 1.6

7.35092 µ 1011

After searching the web, average commute one-way is 16 miles. So we get:

130 000 000 * HH2 * 16L * 1.6L * 365

2.42944 µ 1012

Guesstimates.nb 9

% ê 1.6

1.5184 µ 1012

The answer in the book 2µ 1012 mi.

ü 5.2 Drowning in gasoline

Question: What volume of gasoline does a typical automobile (car, SUV, or pickup) use during its lifetime?Note that this question asks about the lifetime of the vehicle, not the time that you own it. Compare the weightof the fuel to the weight of the car.

H*total number of miles*L

80 * 260 H10^3L êê N

144 222.

H*average miles per gallon*L

6 * 50 êê N

17.3205

%% ê %

8326.66

% * 3.785

31 516.4

My answer 32 m3.

The answer in the book is 20 m3. They use for total number of miles 105 and 20 mpg.

H*gasoline density*L730 kg ë m3

H*total gasoline mass per vehicle*L32 * 730 kg

23 360 kg

My Honda Accord ‘98 weight is º 1300 kg.

ü 5.3 Slowly on the highway

Question: How much total extra time would Americans spend driving each year if we lowered the highwayspeed limit from 65 to 55 mph? (Note that we assume that there is some relationship between posted limits andactual speeds on highways.)Give your answer in lifetimes.

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H* number of employed americans µspeed change ratio µaverage commute time per day per person in hours *L

130 000 000 * H65 ê 55L * H2 * 0.5L êê N

1.53636 µ 108

H* the above result in lifetimes *L% ê H74 * 365 * 24L

237.005

Why using the ratio 65 ê 55 above can seen from this:

H* if average commute time is 1ê2 hour;the distance people have to travel is fixed *LEliminate@81 ê 2 * 65 * x ã s, 1 ê 2 * 55 * y ã s<, 8s<D

11 y ã 13 x

Re-computing with the whole of population of USA:

H* number of americans µspeed change ratio µaverage commute time per day per person in hours *L

300 000 000 * H65 ê 55L * H2 * 0.5L êê N

3.54545 µ 108

H* the above result in lifetimes *L% ê H74 * 365 * 24L

546.935

The book answer is 1000 lifetimes with each life being 100 years.

ü 5.4 Rickshaws and automobiles

Question: What are the relative costs of fuel (per kilometer or per mile) of New York City bicycle rickshaws(human-pedaled taxis) and of automobiles?

ü 5.6 Tire tracks

Question: How far does a car travel before a one-molecule layer of rubber is worn off the tires?

ü 5.7 Working for the car

Question: Your car allows you to travel many miles in just a few hours. However, in addition to the hours youspend driving, you have to spend more hours not driving, hours you spend earning money to pay for your car(eg: depreciation, insurance, fuel). This extra time reduces your average car travel speed. For example, if youdrive 60 miles in one hour and then spend one more hour earning enough money to pay for the driving, thenyour average speed is not 60 mph, but 30 mph.If you add all the time you spend working in order to earn the money to pay for your car to all the time thatyou spend driving your car, what is your average car travel speed?

Guesstimates.nb 11

Question: Your car allows you to travel many miles in just a few hours. However, in addition to the hours youspend driving, you have to spend more hours not driving, hours you spend earning money to pay for your car(eg: depreciation, insurance, fuel). This extra time reduces your average car travel speed. For example, if youdrive 60 miles in one hour and then spend one more hour earning enough money to pay for the driving, thenyour average speed is not 60 mph, but 30 mph.If you add all the time you spend working in order to earn the money to pay for your car to all the time thatyou spend driving your car, what is your average car travel speed?

5. Energy and Work

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