Kata Bognar
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Economics 41
Statistics for Economists
UCLA
Spring 2010
Homework Assignment 3.
 suggested solutions by Miao Sun 
Bayes Theorem.
1. Dan’s Diner employs three dishwasher. Al washes 40% of the dishes and only breaks 1% of those
he handles. Betty and Chuck each wash 30% of the dishes, and Betty breaks only 1% of hers, but
Chuck breaks 3% of the dishes he washes. You go to Dan’s for supper one night and hear a dish
break at the sink. What is the probability that Chuck is on the job?
Answer:
Deﬁne the following events.
•
A
= “Al washes the dishes the night you go for dinner”
,
•
B
= “Betty washes the dishes the night you go for dinner”
,
•
C
= “Chuck washes the dishes the night you go for dinner”
•
BR
= “a dish breaks the night you go for dinner”
.
Then the following probabilities are given in the example:
•
P
(
A
) = 0
.
4
, P
(
B
) = 0
.
3 and
P
(
C
) = 0
.
3
•
P
(
BR

A
) = 0
.
01
, P
(
BR

B
) = 0
.
01 and
P
(
BR

C
) = 0
.
03
You have to ﬁnd the conditional probability
P
(
C

BR
) and the easiest way to do this is to use the
Bayes Theorem.
P
(
C

BR
) =
P
(
C
)
P
(
BR

C
)
P
(
A
)
P
(
BR

A
) +
P
(
B
)
P
(
BR

B
) +
P
(
C
)
P
(
BR

C
)
=
0
.
3
·
0
.
03
0
.
4
·
0
.
01 + 0
.
3
·
0
.
01 + 0
.
3
·
0
.
03
= 0
.
5625
Discrete Distributions.
2.
X
is a random variable with a p.m.f
f
(
x
) =
±
50
x
²
0
.
3
x
0
.
7
50

x
x
= 0
,
1
,
2
,...
50
.
Then,
C
(a)
X
has binomial distribution with the parameters 50 and 0
.
5
.
(b)
X
has a poisson distribution with a parameter 50
.
(c)
X
has binomial distribution with the parameters
50
and
0
.
3
.
(d)
X
has a poisson distribution with a parameter
x.
Explanation:
The probability mass function of a binomial random variable with parameters
n,p
is given by
f
(
x
) =
±
n
x
²
p
x
(1

p
)
n

x
.
The probability mass function of a poisson random variable with a parameter
λ
is given by
f
(
x
) =
e

λ
λ
x
x
!
.
The p.m.f given above is clearly the p.m.f. of a binomial distribution with parameters
n
= 50 and
p
= 0
.
3
.
1
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View Full Document3. According to an article, 35% of American adults have experienced a breakup at least once during
the last 10 years. Of nine randomly selected American adults, ﬁnd the probability that
(a) exactly 5
(b) at most 6
have experienced a breakup at least once during the last 10 years. Let
X
denote the number of
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 Fall '07
 Guggenberger
 Economics, Probability theory, Discrete probability distribution, X1

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