Computer Science 340
Reasoning about Computation
Homework 3
Due at the beginning of class on Wednesday, October 10, 2007
Problem 1
Suppose you toss three fair, mutually independent coins. Define the following events:
•
A  the event that the first coin is heads
•
B  the event that the second is heads
•
C  the event that the third coin is heads
•
D  the event that the number of heads is even
Determine the maximum value of
k
such that these events are
k
wise independent.
Solution:
These events are 3wise independent. First, let us note that these events are not mutually
independent and thus they are not 4wise independent. Indeed,
Pr(
A
∩
B
∩
C
∩
D
) = 0
.
Here, we used the fact that if the events
A
,
B
and
C
happen, then the number of heads is
odd, so the event
D
does not happen. On the other hand
Pr(
A
)
·
Pr(
B
)
·
Pr(
C
)
·
Pr(
D
) =
1
16
,
thus
Pr(
A
∩
B
∩
C
∩
D
) = Pr(
A
)
·
Pr(
B
)
·
Pr(
C
)
·
Pr(
D
)
.
Now we prove that any three of the events
A
,
B
,
C
and
D
are mutually independent.
The events
A
,
B
,
C
are mutually independent by the condition. Let us prove that
A
,
B
,
and
D
are mutually independent. Let
I
A
,
I
B
, and
I
D
be the indicators of the events
A
,
B
and
D
. Fix arbitrary
x
1
, x
2
, x
3
∈ {
0
,
1
}
. Let us compute the following probabilities:
Pr(
I
A
=
x
1
) =
1
2
; Pr(
I
B
=
x
2
) =
1
2
; Pr(
I
D
=
x
3
) =
1
2
;
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Pr(
I
A
=
x
1
&
I
B
=
x
2
&
I
D
=
x
3
)
=
Pr(
I
A
=
x
1
&
I
B
=
x
2
&
I
A
⊕
I
B
⊕
I
C
=
x
3
)
=
Pr(
I
A
=
x
1
&
I
B
=
x
2
&
x
1
⊕
x
2
⊕
I
C
=
x
3
)
=
Pr(
I
A
=
x
1
&
I
B
=
x
2
&
I
C
=
x
1
⊕
x
2
⊕
x
3
)
=
Pr(
I
A
=
x
1
)
·
Pr(
I
B
=
x
2
)
·
Pr(
I
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 Fall '07
 CharikarandChazelle
 Probability theory

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