ECE 310, Spring 2005, HW 3 solutions
Problem E2.41
a) Let Ω be the set of all possible arrangements of
k
indistinguishable photons
in N intervals. By BoseEinstein statistics,
k
Ω
k
=
(
N

1+
k
k
)
. Let
A
be the
set of possible arrangements, at which no two photons occupy the same
interval; in other words, each interval contains either one photon or zero
photons. Then
k
A
k
=
(
N
k
)
and
P
a
=
P
(
A
) =
k
A
k
k
Ω
k
=
(
N
k
)
(
N

1+
k
k
)
b) Let
B
be the event that at least two photons share some unit interval,
then
B
=
A
c
and
P
b
=
P
(
B
) = 1

P
(
A
) = 1

(
N
k
)
(
N

1+
k
k
)
c) Let
C
be the event that two of
k
photons turned to be in the ﬁrst interval
or, in other words,
k

2 remaining photons were arranged in
N

1 intervals.
Hence,
k
C
k
=
(
N

4+
k
k

2
)
and
P
c
=
P
(
C
) =
(
N

4+
k
k

2
)
(
N

1+
k
k
)
Problem E2.42
a) According to BoseEinstein statistics (page 78), there are
m
=
(
10

1+3
3
)
=
(
12
3
)
= 220 distinguishable arrangements.
b) The event
A
corresponds to one of
m
possible arrangements, thus
k
A
k
= 1
and
P
(
A
) =
1
(
12
3
)
= 0
.
0045(45)
c) The event
B
is that none of 3 photons are in the cell
q
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 Spring '05
 HAAS
 Conditional Probability, Probability, Probability theory, possible arrangements, blue pixels

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