Practice Problems for Exam 1
Problem 1.
A random number of bulbs from a certain factory is taken for a test.
The
lifetime in thousands of hours is given by
1
,
20
,
7
,
16
,
22
,
30
,
11
,
2
,
3
,
15
,
12
,
8
,
11
,
20
,
30
,
1
,
5
,
15
.
1. Find the mean and the median.
2. Draw a histogram of the sample.
3. Based on the intervals
[0
,
5)
,
[5
,
10)
,
[10
,
17)
,
[17
,
25)
,
[25
,
35)
, draw the density of the
sample.
Solution.
1.
Mean = 229
/
18 = 22
.
7222
,
Median = 23
/
2 = 11
.
5
.
0
5
10
15
20
25
30
0.5
1.0
1.5
2.0
2.
5
10
15
20
25
30
35
0.01
0.02
0.03
0.04
0.05
3.
Problem 2.
In choosing a lunch a person is faced with three options, salad, meat plate or
pasta. Assume that the person is twice likely to choose pasta than meat or salad. Write the
sample space of the experiment and find the probability that the person chooses meat.
Solution.
The sample space is given by
S
=
{
s, m, p
}
. We know that
P
(
p
) = 2
P
(
m
) =
2
P
(
s
) and on the other hand
P
(
p
) +
P
(
m
) +
P
(
s
) = 1, therefore
P
(
m
) =
P
(
s
) = 1
/
4 and
P
(
p
) = 1
/
2.
Problem 3.
A password consists of a combination of seven digits and letters. How many
different passwords are possible if
1. the first 3 are digits and the other 4 are letters;
1
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2. there are exactly 3 digits and 4 letters;
3. there are at least 2 digits;
4. the first 4 are distinct digits and the rest are either digits or letters;
5. the first 3 are consecutive digits and the rest are different letters;
Solution.
1. 10
3
×
26
4
= 456976000;
2.
(
7
3
)
10
3
26
4
= 15994160000;
3. 36
7

26
7

7
×
10
×
26
6
= 48708249600;
4. 10
×
9
×
8
×
7
×
36
3
= 235146240;
5. 8
×
26
×
25
×
24
×
23 = 2870400.
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 Spring '08
 CITRIN
 Probability, Electromagnet, Probability theory

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