Guesstimates. Some solutions for the book Guesstimation by Weinstein & Adam. Introduction. 1. How to Solve Problems

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1 Guesstimates Some solutions for the book Guesstimation by Weinstein & Adam Anton Antonov Introduction In this notebook (article) I have shown the solutions of some problems from the book Guesstimation: Solving the World s Problems on the Back of a Cocktail Napkin by Lawrence Weinstein and John A. Adam. ( See 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 the geometric mean to estimate averages from lower and upper bounds estimates is adequate in many cases. See the sub-section Geometric mean in the next section. In the first chapter of the book the authors just explain a technique for quick computation of the geometric mean. They do not elaborate much on why it is a better estimate except by giving few examples. (I found their short description inspiring, so I looked for an explanation 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 based on 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 Drowning in 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 the geometric 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 that all models are used equally likely. We have mpg data for 32 car models.

2 2 Guesstimates.nb I have taken the data from R with the command sapply(list(row.names(mtcars),mtcars$mpg),unlist). Some descriptive statistics of the mpg data: Min Max Mean Median StandardDeviation 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 an approximation with a function of the type a + b 1.06 x. The blue line is for the type a + b x.

3 Guesstimates.nb Let us calculate the geometric and arithmetic means based on the minimum and maximum values. H* geometric mean *L gm = 10.4 * H* arithmetic mean *L am = If we place this numbers in the histogram for the mpg distribution it can be easily seen that the arithmetic mean overestimates the average mpg and the geometric mean is a quite good estimate. (Red line for geometric mean, blue line for arithmetic mean.) Let us look at the ratios between the sorted in increasing order mpg s: Min Max Mean Median StandardDeviation 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 *L 10.4 * Mean@ratiosD^ Another way to look at the geometric mean is that it approximates the median value just by using the minimum and maximum values. As an estimator for incomplete and missing data, the median is more robust than

4 4 Guesstimates.nb the mean. In the plot below it can be seen (red grid lines) that the geometric mean is closest to the value of the median of the mpg data I would conjecture that in most cases when we bring minimum and maximum values to estimate a value we do so because we have not build a good intuition and judgment for value s spread and distribution. If the value obeyed a linear law we would have noticed that and would be inclined to use the arithmetic mean with higher certainty. In all other cases we are better off using the geometric mean. This is because, say, life phenomenas tend 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 -- ; -- Long tail -- ; -- 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 the possible 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 the moon (4 10^8 m)? H*thickness of 10 tickets µ 100 million *L H0.001 m ê 10L * 10^ m So it is close to Mount Everest.

5 Guesstimates.nb 5 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? Compare this 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 *L 6.93 * 10^9 * H1.6 *.7 *.5L m^ µ 10 9 m 3 H* world population µ human width µ side *L 6.93 * 10^9 * H.7 *.45L m^ µ 10 9 m 2 H* world population µ area to live *L 6.93 * 10^9 * H2 * 2L m^ µ m 2 ü 3.2 Fore! Question: How many golf balls would it take to circle the Earth at the equator? Golf ball diameter: 0.02 * Earth diameter: ed = * 10^3 * * 10^ * p * ed ê µ 10 8 % ê µ My first un-informed answer is 2 µ golf balls. Earth equatorial radius from WolframAlpha.com is km.

6 6 Guesstimates.nb er = µ 10^ µ 10 6 Several estimates to answer the question: H* Earth's equator lengthêgolf ball diameter *L 2 * p * er ê µ * p * er ê µ * p * er ê µ 10 9 So my answer is 1.5 µ 10 9 golf balls. The answer in the book is 10 9 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? * 2 * 200 * 0.03 * µ 10 8 My answer º1.2 light seconds. The answer in the book is 6 µ 10 8 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 a room, 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 atomic weight in grams. For example, a mole of hydrogen weighs 1 gram and a mole of carbon weighs 12 grams. It is used in chemistry to make sure that there are equivalent numbers of atoms for a chemical reaction.) Compare this to the mass of a mountain, a continent, the moon ( kg), the Earth ( kg). H* Avogadro's constant µ cat mass estimate *L * 10^23 * 0.2 * kg µ kg

7 Guesstimates.nb 7 H* Avogadro's constant µ 4 kg *L * 10^23 * 4 kg µ kg H* my first cat mass estimate *L 0.2 * kg kg The answer in the book is 5 µ 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 * µ H* the result above in tons *L % ê µ 10 8 My answer is 4 µ 10 8 tons. 300 µ 10^6 * 365 * 0.01 * µ 10 9 My answer is m 3. The answers in the book are: -- 3 µ 10 8 m 3 of trash/year; -- According to the US Environmental Protection Agency [9], in 2005 the US generated 245 million (2.45 µ 10 8 ) 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 *L cs = 10. ê 10^

8 8 Guesstimates.nb H* average human body volume ê cell volume *L ê Hcs^3L 6.64 µ H* my body volume ê cell volume *L 0.09 ê Hcs^3L 9. µ The answer in the book is 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 * * 60 êê N The distance above is too little. H* meters run in 10 seconds *L 14 * 60 êê N H* game time in minutes µ 6 µ 10 second run distance *L 90.0 * 6 * % 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 *

9 Guesstimates.nb 9 H*number of nose picking times per day*l 1 * 6 êê N H* world population *L 6.84 * 10^9 H* population µ probability a person to pick his nose during the day *L H6 * 2.5L 6.84 * 10^9 * 15 * µ 10 6 My estimate of the number of people picking their noses at every second is The answer in the book is 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 compare to the circumference of the Earth (2.5 10^4 mi), the distance to the Moon (2.4 10^5 mi), the distance to the Sun (9 10^7 mi), or the distance to Saturn (10^9 mi)? H* number of employed people in USA *L H* number of people µ commute distance estimate in km µ number of days per year *L * J 2 * 120 * 1.6N * µ H* the result above in miles *L % ê µ After searching the web, average commute one-way is 16 miles. So we get: * HH2 * 16L * 1.6L * µ 10 12

10 10 Guesstimates.nb % ê µ The answer in the book 2 µ 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 weight of the fuel to the weight of the car. H*total number of miles*l 80 * 260 H10^3L êê N H*average miles per gallon*l 6 * 50 êê N %% ê % % * My answer 32 m 3. The answer in the book is 20 m 3. They use for total number of miles 10 5 and 20 mpg. H*gasoline density*l 730 kg ë m 3 H*total gasoline mass per vehicle*l 32 * 730 kg 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 highway speed limit from 65 to 55 mph? (Note that we assume that there is some relationship between posted limits and actual speeds on highways.) Give your answer in lifetimes.

11 Guesstimates.nb 11 H* number of employed americans µ speed change ratio µ average commute time per day per person in hours *L * H65 ê 55L * H2 * 0.5L êê N µ 10 8 H* the above result in lifetimes *L % ê H74 * 365 * 24L 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 *L Eliminate@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 µ speed change ratio µ average commute time per day per person in hours *L * H65 ê 55L * H2 * 0.5L êê N µ 10 8 H* the above result in lifetimes *L % ê H74 * 365 * 24L 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 you spend 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 you drive 60 miles in one hour and then spend one more hour earning enough money to pay for the driving, then

12 12 Guesstimates.nb your 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 that you spend driving your car, what is your average car travel speed? 5. Energy and Work