Back of the Envelope
Reasoning
Grad Forum
Tuesday, April 20, 2004
Back of the Envelope Reasoning
• Numeric answers
• Specificity-Resources tradeoff
• Only thing you can do in many domains
– Environmental science, Biophysics, etc.
– New, unfamiliar domains.
• Useful
– Engineers, Scientists, Policymakers, everyone
who reads a newspaper
• Problem solving + feel for numbers
Some examples
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How many K-8 elementary school teachers are in the USA?
How much money is spent on newspapers in USA per year?
How much time would be saved per year nationwide by
increasing the speed limit from 55 to 65 mph?
What is the annual cost of healthcare in USA?
How much tea (weight) is there in China?
Last summer, the US Army bought Microsoft
Windows/Office/Server software for 500,000 computers. The
deal included the software and six years of support. How much
did the army pay for this?
How many K-8 elementary
school teachers are in the USA?
Number of teachers = number of students / students per teacher
Number of students = population * fraction in the age range of K8 students * fraction of kids who go to school
Number of students = 300 mil * 9/75 * 1
Number of teachers = 40 mil / 25 = 1.6 mil
1.9 million, source: Statistical Abstracts.
How much money is spent on
newspapers in USA per year?
Total money spent = Money spent per buyer * number of buyers
Annual expense per buyer = Units bought per year * cost per unit
Annual expense per buyer = 365 * 0.75 = 250
Number of buyers = 300 mil * ¼ = 75 mil
Total money spent = 75 mil * 250 = 20 billion
26 billion, source: Statistical abstracts
How much time would be saved by increasing
the speed limit from 55 to 65 mph?
Total time spent driving = time spent driving per person * number of drivers
Time spent driving per person = Total distance driven per person /Speed
Total distance driven per person = Distance per day * number of days
driving
Number of drivers: assume one driver per household = 1/3 * population =
100 mil
Total distance driven per person = 20 * 365 = 8000
Time spent per person = 8000/55 = 150 hours
Time spent if it was 65 mph = 8000/65 = 120 hours
Time saved per person = 30 hrs
Total time saved = 3 billion hours = 300,000 years.
What is the annual cost of
healthcare in USA?
Lets say everyone was insured.
Total cost of healthcare has to be less than insurance premiums, for
the insurance companies to stay in business.
Cost of healthcare = average insurance premium * population
= 3,000 * 300 million ~ 1 trillion
[1.6 trillion last year. How does this work?]
How much tea is there in China?
Total tea = amount of tea consumed per day *
stockpile in time
Tea consumed per day = tea consumed per
person per cup * number of cups * population
1 billion people
5 cups a day
5 grams a cup
Lets say they stock an years supply
25 billion grams = 25 million kilos per day
In an year 365*25 million ~ 10 billion kilos
The Microsoft Army – How much does
software for 500,000 computers cost?
Total cost on the software = cost of desktop software + server software +
upgrades
Windows XP/Office CDW prices = 650
Cost of desktop software = 500,000 * 650 = 325 mil
Cost of upgrades = 500,000 * 200 = 100 mil
Assume 1 in 100 servers = 5000 servers
Exchange server = 600
Windows 2003 server = 650
SQL Server = 1000
Cost of server software = 2250 * 5000 = 11.25 mil
Cost of upgrades = 400 * 5000 = 2 mil
Total cost = 440 million
Softmart, inc, PA got paid = 470 million
SOLVE
• Suggestions based problem solver
• Represent problem solving progress in an
and/or tree structure
• Incremental solution generation, getnext-solution
• Ordered subgoals
An example
(defSuggestion VolumeFormulaForSphere
(volumeOfObject ?obj ?vol)
:test (shapeOfObject ?obj SphereShape)
:subgoals ((radius ?obj ?radius))
:result-step (evaluate ?vol
(TimesFn 4.187 ;;4/3*Pi
(ExponentFn ?radius 3))))
(annualSales NewspaperCopy
UnitedStatesOfAmerica (YearFn 2003) ?money)
(defSuggestion PerBuyerStrategy
(annualSales ?obj ?place ?time ?money)
:subgoals ((annualExpensePerBuyer ?obj ?place ?time ?money-per-buyer)
(numberOfBuyers ?obj ?place ?time ?number))
:result-step (evaluate ?money (TimesFn ?money-per-buyer ?number)))
(defSuggestion UnitaryStrategyForCost
(annualExpensePerBuyer ?obj ?place ?time ?money-per-buyer)
:subgoals ((annualUnitsBoughtPerBuyer ?obj ?place ?time ?units)
(cost ?obj ?unit-cost))
:result-step (evaluate ?money-per-buyer (TimesFn ?units ?unit-cost)))
(defSuggestion FractionOfTotalStrategy
(numberOfBuyers ?obj ?place ?time ?number)
:subgoals ((populationDuring ?place ?any-time ?total)
(percentOfBuyers ?obj ?place ?percent))
:result-step (evaluate ?number (QuotientFn (TimesFn ?total ?percent) 100)))
(cardinality K-8SchoolTeacher ?numteachers)
(defSuggestion StudentsPerTeacherStrategy
(cardinality K-8SchoolTeacher ?numteachers)
:subgoals ((cardinality K-8SchoolStudent ?numstudents)
(studentsPerTeacher K-8School ?perteacher))
:result-step (evaluate ?numteachers (TimesFun ?numstudents ?perteacher)))
(defSuggestion UniformAgeDistributionStrategy
(cardinality K-8SchoolStudent ?numstudents)
:subgoals ((populationDuring UnitedStatesOfAmerica (YearFn 1997) ?population)
(minimumAge K-8SchoolStudent ?min)
(maximumAge K-8SchoolStudent ?max)
(lifeExpectancyForGroupInRegion UnitedStatesOfAmerica Person
(YearsDuration ?life)))
:result-step (evaluate ?numstudents (TimesFn
(QuotientFn (DifferenceFn ?max ?min) ?life) ?population))))
What next
• Represent 20-30 problems, with the goal
of re-usable representations, in strategies
and theoretical knowledge added.
• Feel for numbers
– Build symbolic representations for numbers
• Large, expensive, upper class, etc.
– Analogical estimator: makes guesses for a
numeric parameter based on analogy.
Numbers in symbolic knowledge
representation
• Consider the Great black-bucked gull
– Wingspan = Large
– Wing-span = 0.272 sq. m.
• Numbers not handled right
– Similarity: computing and making inferences
– Retrieval
– Generalization
CARVE: Symbolizing Quantity
• Don’t do it
– Sorites
– Context/Utterer sensitivity
• Dimensional partitions: Large and Small, based
on distributional properties of the quantity.
(isa Algeria
(HighValueContextualizedFn
Area AfricanCountries))
• Structural partitions: Boiling point and Poverty
line, denote changes of quality.
CARVE
(isa Algeria
(HighValueContextualizedFn
Area AfricanCountries)
.
.
.
Dimensional
partitioning for
each quantity
Ci
Add these facts to
original cases
Quantity 1
S2
S1
L1
S3
Cj
L2
Structural
clustering
using SEQL
Ci*
Analogical Estimator
(GrossDomesticProduct Brazil ?x)
• The value is known.
• Find an analogous case for which value is
known.
• Find anything in the KB which might help
me make an estimate.
Thesis and Evaluation
• More powerful and flexible back of the
envelope reasoning can be done using
these symbolic representations of quantity.
• Evaluation:
– Corpus of problems with/without
representations generated by CARVE.
Numbers in News
• The cost of Mars Pathfinder mission was
270 million.
• 50 billion tax cut.
• 62,000 square feet.