MAT 230 EXAM TWO
1
Directions:
Type
your
solutions
into
this
document
and
be
sure
to
show
all
steps
for
arriving
at
your
solution.
Just
giving
a
final
number
may
not
receive
full
credit.
P
ROBLEM
1
This
question
has
2
parts.
Part
1:
Suppose
that
F
and
X
are
events
from
a
common
sample
space
with
P
(F
)
≠
0
and
P
( )X
≠
0.
(a)
Pro
v
e
that
P
(
X
)
e
=
e
P
(
X
|
F
)
P
(
F
)
+
P
(
X
|
F
¯)
P
(
F
¯
).
Hin
t:
e e
Ex ainpl
wh
y
P
(
X
|
F
)
P
(
F
)
e
=
P
(X
∩
F
)
is
another
way
of
writing
the
definition
of
conditional
probability,
and
then
use
that
with
the
logic
from
the
proof
of
Theorem
4.1.1.
P
(X
|
F
)P
(F
)
=
P
(X
∩
F
)
is
true
because
we
want
to
get
all
probabilities
of
both
events
occurring
but
want
to
avoid
counting
those
events
that
have
already
been
counted.
When
the
intersection
of
the
probabilities
is
subtracted
we
take
away
those
that
have
been
counted
twice.
Since
e
all
e
probabilities
e e
must
e e
add
e e
up
e
to
e
1
e
and
e e
no
e e
more
e e
we
e
can
e e
say
e e
that
e e
the
addition the original probabilities and the complements said probabilities will give of of
a
value
of
1.
(b)
Explain
why
P
(F
|
X)
=
P
(X
F
|
)P
(F
)/P
( )X
is
another
way
of
stating
