Math 361, Problem set 3
Due 9/20/10
1. (1.4.21) Suppose a fair 6sided die is rolled 6 independent times. A match
occurs if side
i
is observed during the
i
th trial,
i
=1
, . . . ,
6.
(a) What is the probability of at least one match during on the 6 rolls.
(b) Extend part (
a
) to a fair
n
sided die with
n
independent rolls. Then
determine the limit of the probability as
n
→∞
.
Answer:
It is much easier to compute the probability that ever roll is
a nonmatch. For any given roll, the probability of a nonmatch is
5
6
.
Since the rolls are independent, the probability that there are no matches
is (5
/
6)
6
. Therefore the probability that there is at least one match is
1

(5
/
6)
6
.
Likewise for any roll in the general case the probability of a nonmatch is
n

1
n
, and hence the probability of at least one match is
1

±
n

1
n
²
n
.
Since lim
n
→∞
(1

1
/n
)
n
=
e

1
, in the limit this is 1

e

1
.
2. (1.4.32) Hunters
A
and
B
shoot at a target; their probabilities of hitting
the target are
p
1
and
p
2
respectively. Assuming independent, can
p
1
and
p
2
be chosen so that
P
(0 hits) =
P
(1 hit) =
P
(2 hits)?
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 Fall '10
 Dr.PaulHorn
 Probability, 4 52 36 52 k

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