16
“mcsftl” — 2010/9/8 — 0:40 — page 431 — #437
Independence
16.1
Definitions
Suppose that we ﬂip two fair coins simultaneously on opposite sides of a room.
Intuitively, the way one coin lands does not affect the way the other coin lands.
The mathematical concept that captures this intuition is called
independence
:
Definition 16.1.1.
Events
A
and
B
are independent if Pr
ŒB�
D
0
or if
±
²
Pr
A
j
B
D
Pr
ŒA�:
(16.1)
In other words,
A
and
B
are independent if knowing that
B
happens does not al
ter the probability that
A
happens, as is the case with ﬂipping two coins on opposite
sides of a room.
16.1.1
Potential Pitfall
Students sometimes get the idea that disjoint events are independent. The
opposite
is true: if
A
\
B
;
, then knowing that
A
happens means you know that
B
D
does not happen. So disjoint events are
never
independent—unless one of them has
probability zero.
16.1.2
Alternative Formulation
Sometimes it is useful to express independence in an alternate form:
Theorem 16.1.2.
A
and
B
are independent if and only if
Pr
ŒA
\
B�
D
Pr
ŒA�
�
Pr
ŒB�:
(16.2)
Proof.
There are two cases to consider depending on whether or not Pr
ŒB�
D
0
.
Case 1
.
Pr
ŒB�
D
0/
:
If Pr
ŒB�
D
0
,
A
and
B
are independent by Definition
16.1.1
.
In addition, Equation
16.2
holds since both sides are 0. Hence, the theorem
is true in this case.
Case 2
.
Pr
ŒB�
>
0/
:
By Definition
15.1.1
,
±
²
Pr
ŒA
\
B�
D
Pr
A
j
B
Pr
ŒB�:
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“mcsftl” — 2010/9/8 — 0:40 — page 432 — #438
Chapter 16
Independence
So Equation
16.2
holds if
±
²
Pr
A
j
B
D
Pr
ŒA�;
which, by Definition
16.1.1
, is true iff
A
and
B
are independent. Hence, the
theorem is true in this case as well.
±
16.2
Independence Is an Assumption
Generally, independence is something that you
assume
in modeling a phenomenon.
For example, consider the experiment of ﬂipping two fair coins. Let
A
be the event
that the first coin comes up heads, and let
B
be the event that the second coin is
heads. If we assume that
A
and
B
are independent, then the probability that both
coins come up heads is:
1
1
1
Pr
ŒA
\
B�
D
Pr
ŒA�
�
Pr
ŒB�
D
2
�
2
D
4
:
In this example, the assumption of independence is reasonable. The result of one
coin toss should have negligible impact on the outcome of the other coin toss. And
if we were to repeat the experiment many times, we would be likely to have
A
\
B
about 1/4 of the time.
There are, of course, many examples of events where assuming independence is
not
justified, For example, let
C
be the event that tomorrow is cloudy and
R
be the
event that tomorrow is rainy. Perhaps Pr
ŒC�
D
1=5
and Pr
ŒR�
D
1=10
in Boston.
If these events were independent, then we could conclude that the probability of a
rainy, cloudy day was quite small:
1
1
1
Pr
ŒR
\
C�
D
Pr
ŒR�
�
Pr
ŒC�
D � D
:
5
10
50
Unfortunately, these events are definitely not independent; in particular, every rainy
day is cloudy. Thus, the probability of a rainy, cloudy day is actually
1=10
.
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 Fall '10
 TomLeighton,Dr.MartenvanDijk
 Computer Science, The Land, Probability theory, mutual independence

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