6.042/18.062J
Mathematics
for
Computer
Science
April
21,
2005
Srini
Devadas
and
Eric
Lehman
Lecture
Notes
Conditional
Probability
Suppose
that
we
pick
a
random
person
in
the
world.
Everyone
has
an
equal
chance
of
being
selected.
Let
A
be
the
event
that
the
person
is
an
MIT
student,
and
let
B
be
the
event
that
the
person
lives
in
Cambridge.
What
are
the
probabilities
of
these
events?
Intuitively,
we’re
picking
a
random
point
in
the
big
ellipse
shown
below
and
asking
how
likely
that
point
is
to
fall
into
region
A
or
B
:
A
B
set of all people
in the world
set of people who
live in Cambridge
set of MIT
students
The
vast
majority
of
people
in
the
world
neither
live
in
Cambridge
nor
are
MIT
students,
so
events
A
and
B
both
have
low
probability.
But
what
is
the
probability
that
a
person
is
an
MIT
student,
given
that
the
person
lives
in
Cambridge?
This
should
be
much
greater—
but
what
it
is
exactly?
What
we’re
asking
for
is
called
a
conditional
probability
;
that
is,
the
probability
that
one
event
happens,
given
that
some
other
event
definitely
happens.
Questions
about
conditional
probabilities
come
up
all
the
time:
• What
is
the
probability
that
it
will
rain
this
afternoon,
given
that
it
is
cloudy
this
morning?
• What
is
the
probability
that
two
rolled
dice
sum
to
10,
given
that
both
are
odd?
• What
is
the
probability
that
I’ll
get
fourofakind
in
Texas
No
Limit
Hold
’Em
Poker,
given
that
I’m
initially
dealt
two
queens?
There
is
a
special
notation
for
conditional
probabilities.
In
general,
Pr
(
A

B
)
denotes
the
probability
of
event
A
,
given
that
event
B
happens.
So,
in
our
example,
Pr (
A

B
)
is
the
probability
that
a
random
person
is
an
MIT
student,
given
that
he
or
she
is
a
Cam
bridge
resident.
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2
Conditional
Probability
How
do
we
compute
Pr
(
A

B
)
?
Since
we
are
given
that
the
person
lives
in
Cambridge,
we
can
forget
about
everyone
in
the
world
who
does
not.
Thus,
all
outcomes
outside
event
B
are
irrelevant.
So,
intuitively,
Pr (
A

B
)
should
be
the
fraction
of
Cambridge
residents
that
are
also
MIT
students;
that
is,
the
answer
should
be
the
probability
that
the
person
is
in
set
A
∩
B
(darkly
shaded)
divided
by
the
probability
that
the
person
is
in
set
B
(lightly
shaded).
This
motivates
the
definition
of
conditional
probability:
Pr
(
A

B
) =
Pr
(
A
∩
B
)
Pr (
B
)
If
Pr
(
B
) = 0
,
then
the
conditional
probability
Pr (
A

B
)
is
undefined.
Probability
is
generally
counterintuitive,
but
conditional
probability
is
the
worst!
Con
ditioning
can
subtly
alter
probabilities
and
produce
unexpected
results
in
randomized
algorithms
and
computer
systems
as
well
as
in
betting
games.
Yet,
the
mathematical
definition
of
conditional
probability
given
above
is
very
simple
and
should
give
you
no
trouble—
provided
you
rely
on
formal
reasoning
and
not
intuition.
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 Spring '11
 Dr.EricLehman
 Computer Science, Conditional Probability, Probability, Probability theory, Probability space

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