6.042/18.062J
Mathematics
for
Computer
Science
November
16,
2010
Tom
Leighton
and
Marten
van
Dijk
Problem
Set
11
Problem
1.
[20
points]
You
are
organizing
a
neighborhood
census
and
instruct
your
census
takers
to
knock
on
doors
and
note
the
sex
of
any
child
that
answers
the
knock.
Assume
that
there
are
two
children
in
a
household
and
that
girls
and
boys
are
equally
likely
to
be
children
and
to
open
the
door.
A
sample
space
for
this
experiment
has
outcomes
that
are
triples
whose
Frst
element
is
either
B
or
G
for
the
sex
of
the
elder
child,
likewise
for
the
second
element
and
the
sex
of
the
younger
child,
and
whose
third
coordinate
is
E
or
Y
indicating
whether
the
e
lder
child
or
y
ounger
child
opened
the
door.
±or
example,
(
B
,
G
,
Y
)
is
the
outcome
that
the
elder
child
is
a
boy,
the
younger
child
is
a
girl,
and
the
girl
opened
the
door.
(a)
[5
pts]
Let
T
be
the
event
that
the
household
has
two
girls,
and
O
be
the
event
that
a
girl
opened
the
door.
List
the
outcomes
in
T
and
O
.
(b)
[5
pts]
What
is
the
probability
Pr
(
T

O
),
that
both
children
are
girls,
given
that
a
girl
opened
the
door?
(c)
[10
pts]
Where
is
the
mistake
in
the
following
argument
for
computing
Pr
(
T

O
)?
If
a
girl
opens
the
door,
then
we
know
that
there
is
at
least
one
girl
in
the
household.
The
probability
that
there
is
at
least
one
girl
is
1
−
Pr
(both
children
are
boys)
=
1
−
(1
/
2
×
1
/
2)
=
3
/
4
.
So,
Pr (
T

there
is
at
least
one
girl
in
the
household)
Pr (
T
∩
there
is
at
least
one
girl
in
the
household)
=
Pr
{
there
is
at
least
one
girl
in
the
household
}
Pr (
T
)
=
Pr
{
there
is
at
least
one
girl
in
the
household
}
=
(1
/
4)
/
(3
/
4)
=
1
/
3
.
Therefore,
given
that
a
girl
opened
the
door,
the
probability
that
there
are
two
girls
in
the
household
is
1/3.
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Problem
Set
11
Problem
2.
[20
points]
Professor
Plum,
Mr.
Green,
and
Miss
Scarlet
are
all
plotting
to
shoot
Colonel
Mustard.
If
one
of
these
three
has
both
an
opportunity
and
the
revolver
,
then
that
person
shoots
Colonel
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 Fall '10
 TomLeighton,Dr.MartenvanDijk
 Computer Science, Probability theory, Colonel Mustard, Professor Leighton

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