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EE 131A
Practice Problems Solutions
Probabilities
Fall 2005
Instructor: Vwani Roychowdhury
1.
P
[
nth is new
] =
m
X
i
=1
P
[
nth is new

nth
=
i
]
p
i
=
m
X
i
=1
P
[
i is not in the first
(
n

1)]
p
i
=
m
X
i
=1
[(1

p
i
)]
n

1
p
i
(a) Let deﬁne:
A
1
=
A
is hit and
A
0
= A is not hit.
P
[
A
0
] =
∞
X
k
=1
P
[
A
0
∩ {
B is hit at the kth time
}
]
=
∞
X
k
=1
P
A
(1

P
B
)[(1

P
A
)(1

P
B
)]
k

1
=
P
A
(1

P
B
)
1

(1

P
B
)(1

P
A
)
(b) Probability that both are hit is equal to:
∞
X
k
=1
[(1

P
A
)(1

P
B
)]
k

1
P
A
P
B
=
P
A
P
B
1

(1

P
A
)(1

P
B
)
(c) the probability that the duel ends after the
n
th round of shots is equal to the
probability that both hit each other exactly at the
n
th round which is equal to:
[(1

P
A
)(1

P
B
)]
n

1
[
P
A
(1

P
B
) +
P
B
(1

P
A
) +
P
A
P
B
]
1
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View Full Document(d) The conditional probability that the duel ends after the
n
th round of shots given
that
A
is not hit is
Prob
[
game ends at n

A isn
0
t hit
] which is equal to:
P
[
game ends at n
∩
A isn
0
t hit
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