**MATH 333, Exam I,
Spring February
14, 2001 NJIT **

** **

**S. Balaji, S. Dhar, M. Naser,
B. Ray**

**I pledge on my honor
that I have abided by the Honor System.**** ___________________**

** **

1. A
professor has two boxes of computer disks: Box C_{1} contains eight
Verbatim disks and three Control Data disks and box C_{2 }contains
three Verbatim disks and five Control Data disks. Given their positions on his
shelf, Box C_{1} has chance 5/12 of being selected, while Box C_{2 }has
chance 7/12 of being selected. A box is selected and a disk is then picked at
random from the selected box.

(a) (10
pts) Compute the probability of V, the event a Verbatim disk is selected.

(b) (8
pts) Given that a Verbatim disk is selected, what is the probability it came
from box C_{2}?

2.
(10 pts) All cars of a particular state are tested for
emission compliance during a year. A particular family has three cars. The
probability that Car 1 will fail the emission test is 0.15, the probability that Car 2 will fail the emission test is
0.08, and the probability that Car 3 will fail the emission test is 0.05.
Assuming the test results for each car are mutually independent, what is the
probability that at most one of the three cars will fail the emission control
test?

3. (10 pts) A person has purchased 8 of 1000 raffle tickets sold in a certain raffle. To determine the 4 prize winners, 4 tickets are to be drawn at random and without replacement. Compute the probability that this person will win at least one prize.

4.
A
certain system can experience three different types of defects. Let A_{i }denote
the probability that the system has a defect of type i, i=1,2,3 and suppose
that P(A_{1})=0.14, P(A_{2})=0.10, P(A_{3})=0.08, _{}, _{}, _{}, _{}.

(a)
(8
pts) Draw a Venn diagram to represent this experiment and shade the portion of
the diagram corresponding to the event {The system has both a type 2 and a type
3 defect but not a type 1 defect}.

(b)
(6
pts) Compute the probability of the event depicted in part (a).

(c)
(8
pts) Compute the probability that the system has exactly two of these defects.

5. A city council consists of 12 members. Three are Republicans, five are Democrats and two are Independents. A committee of three is to be selected.

(a)
(10
pts) Compute the probability of selecting exactly one Democratic committee
member.

(b) (10 pts) The first member selected will be named the committee president, while the second member will be named the committee secretary. Compute the probability that both the president and the secretary will be Democrats.

6. A sample of 20 glass bottles of a particular type was selected and the internal pressure strength of each bottle was determined. Consider the following partial sample information:

Sample median=202.2

Sample
average=190.4

Sample
standard deviation=18.0

Lower
fourth = 25^{th} percentile=196.0

Upper
fourth=75^{th} percentile=216.0

Three
smallest observations: 125.8 188.1
193.7

Three
largest observations: 221.3
230.5 250.2

a. (8
pts) Are there any outliers in the sample? If so, are the outliers mild or
extreme? Justify your answer.

b. (6 pts) Using the information given here,
comment on the symmetry of the distribution of the sample data.

c. (6
pts) Give two different measures of data spread for this sample.

**Brain Teaser **(not
for credit):

You are playing a TV game show
in which you are shown three doors. Behind two of the doors is a goat, while
behind the third door is a new car. You select one door at random. The game
show host then opens one of the unpicked doors to reveal a goat and asks you if
you want to switch your selection to the remaining unselected door. Do you
increase your chance of winning if you switch?