EL630: Homework2 Solutions
Problem 1 solution – the table below lists all the outcomes and their probabilities.
A 1 in
a slot means that switch is closed and a 0 means that switch is open.
outcome
S1
S2
S3
S4
Probability
1
ξ
0
0
0
0
4
)
1
(
p

2
ξ
0
0
0
1
3
)
1
(
p
p

3
ξ
0
0
1
0
3
)
1
(
p
p

4
ξ
0
0
1
1
2
2
)
1
(
p
p

5
ξ
0
1
0
0
3
)
1
(
p
p

6
ξ
0
1
0
1
2
2
)
1
(
p
p

7
ξ
0
1
1
0
2
2
)
1
(
p
p

8
ξ
0
1
1
1
)
1
(
3
p
p

9
ξ
1
0
0
0
3
)
1
(
p
p

10
ξ
1
0
0
1
2
2
)
1
(
p
p

11
ξ
1
0
1
0
2
2
)
1
(
p
p

12
ξ
1
0
1
1
)
1
(
3
p
p

3
ξ
1
1
0
0
2
2
)
1
(
p
p

4
ξ
1
1
0
1
)
1
(
3
p
p

15
ξ
1
1
1
0
)
1
(
3
p
p

16
ξ
1
1
1
1
4
p
Let
=
A
{Signal is received at output},
=
i
S
{i
th
switch is closed}.
a.
We can see that
{
}
16
15
14
13
12
11
10
9
8
7
6
,
,
,
,
,
,
,
,
,
,
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
=
A
.
To find
)
(
A
P
we just add up the probabilities of the outcomes in
A
, to get
4
3
2
3
2
2
3
4
3
2
)
1
(
)
1
(
5
)
1
(
4
)
(
p
p
p
p
p
p
p
p
p
p
p
A
P
+

+
=

+

+

+
=
.
b.
We want to find
)
(
4
A
S
P
, which can be written as
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)
(
)
(
)
(
4
4
A
P
A
S
P
A
S
P
=
.
Now, we can see, from the table above that
{
}
15
13
11
9
7
4
,
,
,
,
ξ
ξ
ξ
ξ
ξ
=
A
S
and the probability of this event is found by adding up the probabilities of the
outcomes in
A
S
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 Spring '11
 voltz
 Probability, Probability theory, switch

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