3113

Name:

1 - Write >uur name in the space provided above.

2- This is a closed book exam.

3- This exam consists of 1 front page + four problems in 7 pages + 1 page with

formula. Please--eheek-your copy and alert the instructor if you are missing

problems or pages. No questions will be allowed after the first 15 minutes.

4- Read each question carefully and neatly write the answers in the space

provided.

5- Show all work. Partial credit will only be given if all work is clearly explained.

For instructor's use only

Problem 1

Problem 2

Problem 3

Problem 4

Total 00

Problem 1. A certain company sends 40% of its overnight mail parcels via express mail service El. Of

these parcels, 2% arrive after the guaranteed delivery time (denote the event "late delivery" by L).

Suppose that 50% of the parcels are sent via express mail service E2 and the remaining 10% are sent via

E3. Of those sent via E2, only 1% arrives late, whereas 5% of the parcels handled by E3 arrive late.

(a) Draw the corresponding tree diagram

' "

(b) If a record of an overnight mailing is randomly selected from the company's file, what is the

probability that the parcel went via El and was late? . v

"

-. ,00%

/

(c) What is the probability that a randomly selected parcel arrived late?

yu . . ~ ?(d) If a randomly selected parcel has arriveo on time, what is the probability that it was not send via El?

^'

t

- \ ,

Problem 2. In a Little League baseball game, team A's pitcher throws a strike 40% of the time and a ball

60% of the time. Successive pitches are independent of one another and the pitcher never hits a batter.

Knowing this, team B's manager has instructed the first batter not to swing at anything. Calculate the

probability that:

(a) The batter walks on the fourth pitch.

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A->"r

(b) The batter walks on the sixth pitch. )^ ^/-^, ( ^ , • (,)

(c) The batter walks.

f*-' es

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(d) The first batter up scores while no one is out (assuming that each batter pursues a no-swing strategy).

A 1

6>U>*

Problem 3. An Engineer has 10 concrete specimens made with siliceous river gravel from Houston and

10 concrete specimens made with limestone from Austin. The Engineer asks the lab personnel to select 15

specimens at random for testing.

(a) What is the p.m.f. of the number of limestone specimens selected for the analysis?

n-x

X - -K

- 10

(b) What is the expected value and standard deviation of the number of limestone specimens selected for

the analysis? \ I /Y \ / M-o \ M / , H \ t A J - / ' ) f\ I \ — \ ^i-' \ N v. N /

x

, I6

V 20- ,X

cr - : 5

(c) What is the probability that all specimens of one of the two types of aggregate are selected for testing?

t

of

o \0 \J

f

\J

|0

5 J No.

20\) What is the probability that the number of limestone specimens is within one standard deviation of its

meanvalue)?

X 1 S -

1° V10? / ( 510 *

10F

| \ o \/

IS

Problem 4. The number of people arriving for treatment at an emergency room can be modeled by a

Poisson process with a rate parameter of five per hour.

(a) What is the probability that exactly four arrivals occur during a particular hour?

\(*jV *<?JL

p (x,M), ^-S —"

(b) What is the probability that at least four people arrive during a particular hour?

-<=

5 2'

(c) How many people do you expect to arrive during A 45-min period?

co

A , , . ,

IfVl ^

ADDITIONAL MATERIAL

SAMPLE VARIANCE

n-\L RANDOM VARIABLE

POISSON RANDOM VARIABLE

x\C RANDOM VARIABLE

= np(\-p]

x n - x M (N-n] M(. M—^7—r -; E(X) = n—; V(X} = \— 1N N N-IJ N N