Math 4653: Elementary Probability: Spring 2007
Homework #3. Problems and Solutions
1. Sec. 2.4: #2:
Find Poisson approximations to the probabilities of the following events in 500
independent trials with probability 0
.
02 of success on each trial:
a)
1 success;
b)
2 or fewer successes;
c)
more than 3 successes.
Solution.
Let
X
be the number of successes. The actual distribution of
X
is a binomial distribution
with
n
= 500 and
p
= 0
.
02.
Therefore,
μ
=
np
= 10, and the Poisson distribution provides a good
approximation.
a
)
P
(
X
= 1)
≈
e

10
10
1
1!
≈
0
.
0004540
.
b
)
P
(
X
≤
2)
≈
2
k
=0
e

10
10
k
k
!
=
e

10
(1 + 10 + 50)
≈
0
.
0027694
.
c
)
P
(
X >
3)
≈
1

3
k
=0
e

10
10
k
k
!
= 1

e

10
1 + 10 + 50 +
500
3
≈
0
.
98966
.
2. Sec. 2.4: #6a):
A box contains 1000 balls, of which 2 are black and the rest are white. Which
of the following is most likely to occur in 1000 draws with replacement from the box?
fewer than 2 black balls,
exactly 2 black balls,
more than 2 black balls.
Solution.
Let
X
be the number of black balls drawn. The actual distribution of
X
is a binomial
distribution with
n
= 1000 and
p
= 0
.
002.
Therefore,
μ
=
np
= 2 , and the Poisson distribution
provides a good approximation. We compute:
P
(
X <
2)
≈
1
k
=0
e

2
2
k
k
!
=
e

2
(1 + 2)
≈
0
.
4060
,
P
(
X
= 2)
≈
e

2
2
2
2!
=
e

2
·
2
≈
0
.
2707
,
P
(
X >
2)
=
1

P
(
X <
2)

P
(
X
= 2)
≈
1

0
.
4060

0
.
2707 = 0
.
3233
.
Therefore, of the three events, the most likely is
“fewer than
2
black balls”
.
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 Fall '03
 aldous
 Probability, Binomial distribution, 1 K, 2 2k, 1  k, 8 8 8 j

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