One of the most important thing, if not the most important, in the university
the cultivation of the ability to extrapolate. Obviously we can not anticipate
all the problems you are going to encounter in the future and prepare you for them.
So what we try to do is to teach you the essentials and hope that you can deal with
deviations or new problems successfully.
Here several examples of extrapolation are going to be given to illustrate
what I mean by "extrapolation" and how to deal with new problems with the knowledge
and experiences you presently have.
The following problems are essentially a variation of the TV viewers survey problem
of Example 1,
and can be solved by basically the same approach.
Problem: A survey of TV viewers shows the following results:
To the question "Do you watch comedies ?", 374 replied "Yes".
To the question "Do you watch sports ?", 360 replied "Yes".
To the question "Do you watch detective stories ?", 350 replied "Yes".
To the question "Do you watch comedies and sports ?", 134 replied "Yes".
To the question "Do you watch comedies and detectives ?", 96 replied "Yes".
To the question "Do you watch detectives and sports ?", 241 replied "Yes".
To the question "Do you watch all three ?", 37 replied "Yes".
Find the percentages of people who watch only comedies, only sports, only detective stories,
comedies and sports, comedies and detectives, sports and detectives, and all three.
By doing the same kind of analyses on the unknowns and data, one can find these
The total number of people who watch at least one of these programs is
374 + 360 + 350 - 134 - 96 - 241 + 37 = 650,
because each of 134, 96 and 241 is counted twice into 374 + 360 + 350,
and 37 is counted in three times and subtracted out three times in
374 + 360 + 350 - 134 - 96 - 241.
Similarly the number of those watching only comedies is
374 - 134 - 96 + 37 = 181,
the number of those watching only detectives is
350 - 241 - 96 + 37 = 50,
the number of those watching only sports is
360 - 134 - 241 + 37 = 22,
the number of those watching only comedies and detectives is
96 - 37 = 59,
the number of those watching only comedies and sports is
134 - 37 = 97,
the number of those watching only detectives and sports is
241 - 37 = 204.
The calculation of the percentages are omitted.
300 people were surveyed on TV programs they watch and all 300 responded. 75 of them say they watch
a regular sports program,
210 say they watch a regular comedy program, and 36 say they watch both. There is a
special detective story program which conflicts with the regular
sports and comedy programs. If those who watch regular programs do not
want to miss them, how many people can watch the special ?
To answer the question, all you have to do is to find the total number of
people who watch at least one of the regular sports and comedy programs.
Then 300 minus that number is the answer. Thus this problem is essentially
the same as
and the answer is 300 - (75 + 210 - 36) = 51.
300 people were surveyed. 50 of them watch sports,
140 watch comedies, and 134 do not watch either of them.
Then how many of them watch both comedies and sports ?
The relationships among various groups of people in this problem
are the same as those of
Only the data are slightly different.
Since 300 were surveyed and 134 do not watch either sports or comedies,
166 watch at least one. Hence 50 + 140 - 166 = 24 watch both.
Thus 50 - 24 = 26 watch only sports, and 140 - 24 = 116 watch only comedies.
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