Tutorial 1

SQQS1043 Probability and Statistics

lyf – Mar’13

Second Semester 2012/2013

Tutorial Chapter 1

1. Decide whether each distribution is a probability distribution. Give a reason to your

answer.

a.

U -2 0 2 4

P(U) 0.15 0.45 0.22 0.18

b.

V 1 2 3 4

P(V) 0.4 0.2 0.6 -0.2

c.

X 0 1 2 3

P(X) 315 31

13 3112 31

2

d.

Y 5 10 15 20

P(Y) 94 9

1 92 9

2

2. A box contains 12 marbles of which 4 are blue and the others are red. A random sample

of three pens is taken. Let X denotes the number of red pen in the sample.

a. Construct a tree diagram to illustrate the event.

b. Develop the probability distribution of X.

c. Find the probability that

i. Exactly one marble is red.

ii. At most two marbles are red.

iii. At least two marbles are blue.

d. Find the expected number of red marbles in the sample.

3. A discrete random variable X has the following probability distribution.

X 1 2 3 4

P(X) 3c 4c 5c 6c

a. Find the value of the constant c.

b. Find E(X) and standard deviation of X.

c. Find P(X > E(X)).

d. Find the moment generating function of X.


lyf – Mar’13

4. BCom Mobile One manufactures hand phone parts that are supplied to many hand phone

companies. The Quality Control Departments check every part for defects before it is

shipped to another company but a few defective parts still pass through these inspections

undetected. Let X be the number of defective hand phone parts. The following table lists

the frequency distribution of X for the past 80 shipments.

X 1 2 3 4 5

Frequency 8 24 20 16 12

a. Construct a probability distribution for the number of defective hand phone parts.

b. Represent graphically the probability distribution for above.

c. Construct the cumulative distribution for the number of defective hand phone parts.

d. Find the probability that the number of defective hand phone parts is

i. At most four

ii. Exact three

e. Find the expected number and standard deviation of defective hand phone parts per

shipment.

5. A discrete random variable X has the following probability distribution.

X 0 1 2 3 4

P(X) 0.26 p q 0.05 0.09

a. Write down an equation satisfied by p and q.

b. Find E(X) in terms of p and q.

c. Given that E(X) = 1.56, find the value of p and q.

d. Find the standard deviation of X.

6. Suppose a random variable W has the following cumulative distribution function.

51

54,77.0

43,64.0

32,25.0

21,11.0

1,0

)(F

W

W

W

W

W

W

W

a. Find )3(P W .

b. Find )4(P W .

c. Find )2(P W .

d. Find the distribution of W.

e. Draw a bar chart for the probability distribution.


lyf – Mar’13

7. According to a survey, it is estimated that 75% of the students entering a particular

college will graduate in four years. Random samples of three students are selected from

this college. Let X denotes the number of students in this sample who will graduate in

four years.

a. Construct the probability distribution table of X.

b. Find the probability that less than two students will graduate in four years.

c. What is the probability that at least two students will graduates in four years?

8. Figure below shows the probability of the number of months required to complete the

small construction projects.

Based on figure above;

a. Find the value of a.

b. Calculate the expected value and standard deviation for the number of months

required to complete the projects.

c. Construct the cumulative probability distribution.

9. A shipment of 7 imported cars contains three two-seater sports cars. A car rental

company makes a random purchase of 3 of the cars.

a. Let Y be a random variable represent the number of two-seater sport car purchased by

the car rental company. Find the probability distribution of Y.

b. Find

i. P(Y 2)

ii. P(1 < Y 3)

10. The Student Affair Department in UUM received 4 calls every 20 minutes. Based on the

time given, find the probability that the department will received:

a. Not more than 3 calls.

b. Between 3 and 6 calls in 45 minutes.

c. Less than 8 calls in 100 minutes.

3a

0.2

a

P(Y)

)

1 2

Time (months)

3 4


lyf – Mar’13

11. A university found that 20% of its students withdraw without completing the

introductory statistics course. Assume that 20 students registered for the course this

quarter. Compute:

a. The probability that two or fewer will withdraw.

b. The probability that from two to 4 will withdraw.

c. The probability that more than 9 will not withdraw.

d. The expected number of withdraw.

12. In a book exhibition last year, Kerang Book’s company made a mega sale on their books.

The marketing manager found that probability a person will buy their books is 0.45. If

there are 10 customers in a certain time, what is the probability of the customers buying

the books

a. Only 3 person?

b. Between 4 and 8 person?

c. Not more than 7 person?

d. From 6 to 8 person?

13. a. State two conditions which must be satisfied for a situation to be modeled by a

binomial distribution.

b. Single cards, chosen at random, are given away with bars of chocolate. Each card

shows a picture of one of 20 different football players. Richard needs just one picture

to complete his collection. He buys 5 bars of chocolate and looks at all the pictures.

Find the probability that

i. Richard does not complete his collection.

ii. He has the required picture exactly once.

14. A researcher is doing the study about traffic at Melor road. He found that about 180 cars

were using that road in an hour. What is the probability of cars using the road

a. Only 2 in a minute.

b. Less than 4 in 3 minutes.

c. Between 2 and 5 in a minute.

15. According to the Journal of Business, 27% of a business owned by non-Hispanic whites

nationwide is women-owned firms.

a. In a sample of 200 small business owned by non-Hispanic whites, how many would

you expect to be female owned?

b. If eight small business owned by non-Hispanic whites were randomly selected, what

is the probability that none are female owned? What is the probability that half are

female owned?

16. Twelve six-sided fair dice are tossed simultaneously. What is the probability that each

odd number appears exactly two times and that each even number appears exactly two

times?


lyf – Mar’13

17. U.S. airlines average about 1.2 fatalities per month. Assume the probability distribution

for the number of fatalities per month can be approximated by a Poisson distribution,

what;

a. The probability that no fatalities will occur during any given month?

b. The probability that three fatalities will occur during two months?

c. Is the mean and standard deviation of the number of fatalities per month?

18. 5% of air conditioner produced by a manufacturing company is spoiled. Let we have 12

air conditioner from that company. What is the probability to get

a. One air conditioner spoiled?

b. Between 5 and 8 air conditioner spoiled?

c. More than 7 air conditioner not spoiled?

19. a. Computer breakdowns occur randomly on average once every 48 hours of use.

Calculate the probability that there will be fewer than 4 breakdowns in 60 hours of

use.

b. The number of emergency telephone calls to the electricity board office in a certain

area in t minutes is known to follow a Poisson distribution with mean t801 . Find the

probability that there will be at least 3 emergency telephone calls to the office in any

20-minute period.

20. The caps on soda bottles are examined with a scanning device in order to determine if

they properly set. Experience dictates the probability of detecting an improperly set cap

is 0.01.

a. What is the probability that the first improperly set cap will be detected on the tenth

bottle?

b. Find the mean and the variance of the number of bottles examined until the next

improperly set cap is found.

21. Suppose that in a certain locale 30% of the households purchase Brand A soap powder,

50% purchase Brand B soap powder, and 20% of the households use Brand C soap

powder. For a random sample of n = 10 households selected with replacement, what is

the probability that six use Brand A, three use Brand B, and one uses Brand C?

22. For a particular variety of electronic bulletin board the number of components having

unacceptable reliability coefficients was evenly distributed between 7 and 30.

a. Determine the probability of finding at least 15 unacceptable components.

b. What is the average number of acceptable components?

c. What is the standard deviation?

23. Suppose that out of 50 property owners from a given city it is known that 30 support a

bond issue for the addition of a new wing to the public library but 20 do not. If five

property owners are selected at random at a town meeting, what is the probability that

more than one of them will favour the bond issue?


lyf – Mar’13

24. Suppose that a basketball player has a 75% of chance of making a free throw shot. The

player shoots free throws until a total of 10 are made.

a. What is the probability that 12 shots will be required in order to sink 10 of them?

b. What is the probability that it will take 15 shots in order to sink 10 of them?

c. If X is the minimum number of free throws needed to sink of them, find E(X) and

V(X).

25. Suppose an individual is going to roll a bowling ball until he or she get a strike. If, under

independent rolls, this person has a probability of 0.20 of making a strike and if X is the

number of rolls it takes to make the first strike,

a. Determine the probability that fewer than five rolls will be needed to make the first

strike.

b. What is the average number of rolls that the person needs to make to get the first

strike?

26. A fair coin is tossed a larger number of times.

a. What is the probability that seventh head is obtained on the tenth toss?

b. What is the probability that it is obtained on the fifteenth toss?

27. Seated at a table are five individuals, four of which are registered Democrats.

a. Let the random variable X be the number of non-Democrats selected. If two

individuals are selected at random without replacement, what is the probability that

the non-Democrat will be one of those selected?

b. What is the probability that two Democrats will be selected?

28. A box contains 300 discs of different colours. There are 100 pink discs, 100 blue discs

and 100 orange discs. The discs of each colour are numbered from 0 to 99. Five discs are

selected at random, one at a time, with replacement. Find

a. The probability that no orange discs are selected.

b. The probability that exactly 2 discs with numbers ending in a 6 are selected.

c. The probability that exactly 2 orange discs with numbers ending in a 6 are selected.

d. The mean and variance of the number of the pink discs selected.

29. For each of the following moment generating functions, find the associated probability

distribution. Then, determine the mean and variance of X.

a. ( ) ( )

b. ( ) ( )

c. ( ) ( )

d. ( )

Tutorial 1

Documents

Transcript of Tutorial 1