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

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

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

1. A professor has two boxes of computer disks: Box C₁ contains seven Verbatim disks and three Control Data disks and box C₂ contains two Verbatim disks and eight Control Data disks. Given their positions on his shelf, Box C₁ has chance 5/8 of being selected, while Box C₂ has chance 3/8 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 C₂?

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 0.05, the probability that Car 2 will fail the emission test is 0.10, and the probability that Car 3 will fail the emission test is 0.02. Assuming the test results for each car are mutually independent, what is the probability that at least two of the three cars will fail the emission control test?

3. (10 pts) A person has purchased 10 of 1000 raffle tickets sold in a certain raffle. To determine the 5 prize winners, 5 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(A₁)=0.12, P(A₂)=0.07, P(A₃)=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 1 and a type 2 defect but not a type 3 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 10 members. Four are Republicans, three are Democrats and three 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 Republicans.

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?