18.05
Lecture
5
February
14,
2005
§
2.2
Independence
of
events.
P
(
A B
)
=
P
(
AB
)
P
(
B
)
;

Definition
 A
and
B
are
independent
if
P
(
A B
)
=
P
(
A
)

P
(
AB
)
P
(
A B
)
=
=
P
(
A
)
P
(
AB
)
=
P
(
A
)
P
(
B
)

P
(
B
)
Experiments
can
be
physically
independent
(roll
1
die,
then
roll
another
die),
or
seem
physically
related
and
still
be
independent.
Example:
A
P
(
A
1
)
=
{
}
.
P
(
B
2
{
1
,
3
}
=
P
(
AB
1
3
odd ,
B
)
=
)
=
,
therefore
independent.
=
=
{
1,
2,
3,
4
}
.
Related
events,
but
independent.
.AB
=
2
3
2
P
(
AB
)
=
1
3
2
×
Independence
does
not
imply
that
the
sets
do
not
intersect.
Disjoint
=
Independent.
If
A,
B
are
independent,
find
P
(
AB
c
)
P
(
AB
)
=
P
(
A
)
P
(
B
)
AB
c
=
A
\
AB
,
as
shown:
so,
P
(
AB
c
)
=
P
(
A
)

P
(
AB
)
=
P
(
A
)

P
(
A
)
P
(
B
)
=
P
(
A
)(1

P
(
B
))
=
P
(
A
)
P
(
B
c
)
therefore,
A
and
B
c
are
independent
as
well.
similarly,
A
c
and
B
c
are
independent.
See
Pset
3
for
proof.
Independence
allows
you
to
find
P
(intersection)
through
simple
multiplication.
14
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2
Example:
Toss
an
unfair
coin
twice,
these
are
independent
events.
P
(
H
)
=
p,
0
≤
p
≤
1
,
find
P
(“
T
H
)
=
tails
first,
heads
second
P
(“
T
H
)
=
P
(
T
)
P
(
H
)
=
(1

p
)
p
1
Since
this
is
an
unfair
coin,
the
probability
is
not
just
4
T H
1
=
If
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 Spring '11
 MAtuka
 Probability theory, unfair coin

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