6.042/18.062J Mathematics for Computer Science
December 17, 2008
Tom Leighton and Marten van Dijk
Final Exam
Name:
• This ﬁnal is
closed book
, but you may have three 8.5”
×
11” sheets with notes in
your own handwriting on both sides.
• You may not use a calculator or a Python interpreter, and while exercising skill with
a PostScript interpreter would highly impress at least one member of the course
staff, you are not allowed to use that either. You
may
work with any of the follow
ing: a slide rule, an abacus, a Curta, Napier’s bones, any original version of the
Antikythera mechanism, an Enigma machine, and/or the difference engine. You
are also permitted to use a perpetual motion machine as a source of energy, and an
antigravity device to elevate yourself above the rest of the class.
• You may assume all of the results presented in class.
• Please show your work. Partial credit cannot be given for a wrong answer if your
work isn’t shown.
• Write your solutions in the space provided. If you need more space, write on the
back of the sheet containing the problem. Please keep your entire answer to a prob
lem on that problem’s page.
• Be neat and write legibly. You will be graded not only on the correctness of your
answers, but also on the clarity with which you express them.
• If you get stuck on a problem, move on to others. The problems are not arranged in
order of difﬁculty.
• Please resist the urge to roll on the ﬂoor laughing out loud.
• You have three hours to complete the exam.
Problem
1
2
3
4
5
6
7
8
9
10
Total
Points
25
25
25
15
15
15
25
15
20
20
200
Score
Grader
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Final
Exam
Problem 1. [25 points] The Final Breakdown
Suppose the 6.042 ﬁnal consists of:
• 36 true/false questions worth 1 point each.
• 1 induction problem worth 15 points.
• 1 giant problem that combines everything from the semester, worth 49 points.
Grading goes as follows:
• The TAs choose to grade the easy true/false questions. For each individual point,
they ﬂip a fair coin. If it comes up heads, the student gets the point.
• Marten and Brooke split the task of grading the induction problem.
–
With 1/3 probability, Marten grades the problem. His grading policy is as
follows: Either he gets exasperated by the improper use of math symbols and
gives 0 points (which happens with 2/5 probability), or he ﬁnds the answer
satisfactory and gives 15 points (which happens with 3/5 probability).
–
With 2/3 probability, Brooke grades the problem. Her grading policy is as
follows: She selects a random integer point value from the range from 0 to 15,
inclusive, with uniform probability.
• Finally, Tom grades the giant problem. He rolls two fair
seven
sided dice (which
have values from 1 to 7, inclusive), takes their product, and subtracts it from 49 to
determine the score. (Example: Tom rolls a 3 and a 4. The score is then 49
−
3
4
=
·
37.)
Assume all random choices during the grading process are mutually independent.
The problem parts start on the next page. Show your work to receive partial credit.
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 Fall '10
 TomLeighton,Dr.MartenvanDijk
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