IBS – Quantitative Methods

Probability – Assignment, Additional Questions

1. Two dice are thrown, what is the probability of getting the sum being 8 or the sum being
10?
2. Two dice are thrown simultaneously. Find the probability that the sum being 6 or same
number on both dice.
3. Suppose a coin is flipped 3 times. What is the probability of getting two tails and one
head.
4. Two persons A and B appeared for an interview for a job. The probability of selection of
A is 1/3 and that of B is ½. Find the probability that
a. Both of them will be selected b. Only one of them is selected c. None of
them is selected
5. A stockist has 20 items in a lot. Out of which 12 are non-defective and 8 are defective. A
customer selects from the lot. What is the probability that out of these three items i) three
items are defective ii) two are non-defective and one is defective.
6. A test paper containing 10 problems is given to three students A, B , C. It is considered
that student A can solve 30% problems. Find the probability that the problem chosen
from the test paper will be solved by all the three students
7. From a pack of 52 cards,2 cards are drawn at random. Find the probability that one is
king and other is queen.
8. An urn contains 4 black balls. If 3 balls are drawn at random, find the probability that (i)
all are black ii) all are white iii) one white and 2 black.
9. A box containing 5 green balls and 3 red color balls. Find the probability of selecting 3
green color balls one by one i) without replacement ii) with replacement
10. If P(AUB) =0.3, P(A) =0.6, P(B)=0.7. Find the value of P(B/A) and P(A/B)
11. In a certain town, males and females form 50% of the population. It is known that 20% of
the males and 5% of the females are unemployed. A research student studying the
employment situation selects unemployed persons at random. What is the probability that
the person selected is (i) a male (ii) a female?
12. A business firm has invited applications for managerial post. The probability that an
applicant has a postgraduate qualification is 0.3 and that he has work experience is 0.7,
and that he both postgraduate qualification and work experience is 0.4. Assuming that 50
persons have applied for this managerial post in the company. Find out how many
applicants would have either a postgraduate degree or adequate work experience.
13. Five men in a company of 20 are graduates. If 3 men are picked out of the 20 at random,
what is the probability that they are all graduates? What is the probability of at least one
graduate?
14. A bag contains 5 white and 8 red balls. Two drawings of 3 balls are made such that (a)
the balls are replaced before the second trial and (b) the balls are not replaced before the
 second trial . Find the probability that the first drawing will give 3 white and the second,
3 red balls in each case.
15. From a sales force of 150 persons, one will be selected to attend a special sales meeting.
If 52 of them are unmarried, 72 are college graduates and ¾ of the 52 that are unmarried
are college graduates, find the probability that the sales person selected at random will ne
neither single nor a college graduate,
16. An MBA applies for a job in two firms X and Y. The probability of his being selected in
firm X is 0.7 and being rejected at Y is 0.5. The probability that he will selected by one
of the firms?
17. The probability that a new marketing approach will be successful is 0.6. The probability
that the expenditure for developing the approach can be kept within the original budget is
0.5. The probability that both of these objectives is 0.30. What is the probability that at
least one of these objectives will be achieved. For the two events described above,
determine whether the events are independent or dependent.
18. The probability that a trainee will remain with a company is 0.6. The probability that an
employee earns more than Rs 10,000 per month is 0.5. The probability that an employee
who is a trainee remained with the company or who earns more than Rs 10,000 per
month is 0.7. What is the probability that an employee earns more than Rs 10,000 per
month given that he is trainee who stayed with the company?
19. A market survey was conducted in four cities to find out the preference for a brand A
soap. The responses are shown below:
Delhi Kolkota Chennai Mumbai
Yes 45 55 60 50
No 35 45 35 45
No Opinion 5 5 5 5
(a) What is the probability that a consumer selected at random at random, preferred
brand A?
(b) What is the probability that a consumer preferred brand A and was from Chennai?
(c) What is the probability that a consumer preferred brand A, given that he was from
Chennai?
(d) Given that a consumer preferred brand A, what is the probability that he was from
Mumbai?
20. In a factory manufacturing pens, machines X, Y and Z manufacture 30, 30 and 40 percent
of the total production of pens, respectively. Of their output 4,5,and 10% of the pens
respectively. Of their output 4,5,and 10 percent of the pens are defective. If one pen is
selected at random, and it is found to be defective, what is the probability that it is
manufactured by machine Z?
NORMAL DISTRIBUTION
Normal Distribution: Is a continuous
probability distribution in which the curve is
bell-shaped having a single peak. The mean
of the distribution lies at the centre of the
curve.
The normal probability distribution is given
by
1
.
√2
Standard Normal Probability Distribution:
or x = µ + Z
Ex: i. Indicate the area in a normal curve when z = + 1.75
ii. Locate the area in a normal curve when Z = - 1.96
iii. Show the area in a normal curve which is more than Z = 1.5
iv. Find the area in a normal curve to the left of Z at -1.60
v. State the area in a normal curve enclosed between + 1.25 to -1.25
vi. Find the area in a normal curve to the left of Z at 1.45 and to the left of Z at -1.25
vii. Find P (-1.96 ≤ Z ≤ + 1.96)
Problems
1. 1000 light bulbs with a mean life of 120 days are installed in a new factory and their length of
life is normally distributed with standard deviation of 20 days
a) How many bulbs will expire in less than 90 days
b) If it is decided to replace all the bulbs together, what interval should be allowed between
replacements if not more than 10% should expire before replacement?
2. The lifetimes of certain kinds of electronic devices have a mean of 300 hours and standard
deviation of 25 hrs. Assuming that the distribution of these lifetimes which are measured to the
nearest hour, can be approximated closely with a normal curve
a) Find the probability that any one of these electronic devices will have lifetime of more than
350 hrs
b) What percentage will have lifetimes of 300 hrs or less
c) What percentage will have lifetimes from 220 or 260 hrs
3. Assume that the test scores from a college admissions test are normally distributed with a
mean of 450 and a standard deviation of 100
a) What percentage of people taking the test score are between 400 and 500?
b) Suppose someone received a score of 630. What percentage of the people taking the test
score
better? What percentage score more?
c) If a particular university will not admit any one scoring below 480, what percentage of the
person
 taking the test would be acceptable to the university
4. In an intelligence test administered to 1000 students, the average score was 42 and standard
deviation 24. Find (a) the number of students exceeding a score of 50 (b) the number of students
exceeding a score of 50 c) the value of the score exceeded by the top 100 students.
5. A sample of 100 dry battery cells tested to find the length of life produced the following
results:
Mean = 12 hrs, σ = 3 hrs, Assuming that the data are normally distributed, what percentage of
battery cells are expected to have life: (i) more than 15 hrs (ii) less than 6 hrs (iii) between 10
and 14 hrs
6. The monthly income of 1000 employees are normally distributed around a mean of Rs 2,500
with a standard deviation of Rs 250. Find the number of employees whose monthly income
would be
i) between Rs 2,000 and Rs 3,000 ii) Less than Rs 2,000 and iii) More than Rs 3,000
7. A Sales Tax Officer has reported that the average sales of the 500 businesses that he has to
deal with during a year amount to Rs 36,000 with a standard deviation of Rs 10,000. Assuming
that the sales in these businesses are normally distributed, find
i) The number of businesses the sales of which are over Rs 40,000
ii) The Percentage of businesses, the sales of which are likely to range between Rs 30,000 and
Rs 40,000
iii) The probability of Area under the normal curve
Proportions of Area under the normal curve
Z: 0.25 0.40 0.50 0.60
Area: 0.0987 0.1554 0.1915 0.2257
8. A sample of 100 dry battery cells tested to find the length of life produced the following
results: µ = 12 hrs, σ = 3 hrs. Assuming that the data are normally distributed. What percentage
of battery cells are expected to have life. I) more than 15 hrs ii) less than 6 hrs iii) Between 10
and 14 hrs
Given Z 2.5 2.0 1.0 0.67
Area 0.4938 0.4772 0.3413 0.2487
9. In an intelligence test administered to 500 students, the average score was 42 and standard
deviation was 24. Find
a) The number of students whose score exceeded 50
b) The number of students who got a score between 30 and 40
c) The number of students who got a score above 60