Computer Science 340
Reasoning about Computation
Homework 6 (take home midterm)
Due at the beginning of class on Wednesday, November 7, 2007
There are six problems on the midterm.
You must work on them on your own. No collaboration is allowed.
Problem 1
Consider a random subset
R
of
{
1
,
2
, . . . , n
}
created as follows: Initially,
R
is empty.
For each
i
∈ {
1
,
2
, . . . , n
}
,
i
is included in
R
with probability
p
. Here
p
∈
[0
,
1] is a given
parameter, and these choices are mutually independent.
(a)
Compute the probability that
R
contains
k
elements.
(b)
Compute the probability that
R
contains an odd number of elements.
Note:
For full credit, you should give a closed form expression for this probability. A
summation is not considered a closed form expression.
Hint:
If
a
is the probability that
R
contains an odd nmber of elements, let
b
be the prob
ability that
R
contains an even number of elements. Find a relation between
a
and
b
other
than the trivial
a
+
b
= 1.
Problem 2
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 Fall '07
 CharikarandChazelle
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