Massachusetts Institute of Technology
6.042J/18.062J, Spring ’10
: Mathematics for Computer Science
May 3
Prof. Albert R. Meyer
revised May 3, 2010, 616 minutes
Solutions
to
InClass
Problems
Week
13,
Mon.
Problem
1.
Suppose there is a system with
n
components, and we know from past experience that any partic
ular component will fail in a given year with probability
p
. That is, letting
F
i
be the event that the
i
th component fails within one year, we have
Pr
{
F
i
}
=
p
for
1
≤
i
≤
n
. The
system
will fail if
any
one
of its components fails. What can we say about the
probability that the system will fail within one year?
Let
F
be the event that the system fails within one year. Without any additional assumptions,
we can’t get an exact answer for Pr
{
F
}
. However, we can give useful upper and lower bounds,
namely,
p
≤
Pr
{
F
} ≤
np.
(1)
We may as well assume
p <
1
/n
, since the upper bound is trivial otherwise. For example, if
n
=
100
and
p
=
10
−
5
, we conclude that there is at most one chance in 1000 of system failure
within a year and at least one chance in 100,000.
Let’s model this situation with the sample space
S
::=
P
(
{
1
,...,n
}
)
whose outcomes are subsets of
positive integers
≤
n
, where
s
∈ S
corresponds to the indices of exactly those components that fail
within one year. For example,
{
2
,
5
}
is the outcome that the second and ﬁfth components failed
within a year and none of the other components failed. So the outcome that the system did not
fail corresponds to the emptyset,
∅
.
(a)
Show that the probability that the system fails could be as small as
p
by describing appropriate
probabilities for the outcomes. Make sure to verify that the sum of your outcome probabilities is
1.
Solution.
There could be a probability
p
of system failure if all the individual failures occur to
gether. That is, let Pr
{{
1
,...,n
}}
::=
p
, Pr
{∅}
::=1
−
p
, and let the probability of all other outcomes
be zero. So
F
i
=
{
s
∈ S

i
∈
s
}
and Pr
{
F
i
}
=
0+0+
···
+0+
Pr
{{
1
,...,n
}}
=
Pr
{{
1
,...,n
}}
=
p
.
Also, the only outcome with positive probability in
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 Spring '11
 Prof.AlbertR.Meyer
 Computer Science

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