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STATISTICS FOR MANAGEMENT
PROBLEMS ON PROBABILITY AND DISTRIBUTIONS
1. In a factory there are two machines producing 30% and 70%
respectively of the total output. Out
of the items produced by the first machine 4% are defectives,
whereas out of the items produced by
the second machine 6% are defectives. An item is drawn at random
from the production line and it
is found to be defective. What is the conditional probability
that this item was produced by the
second machine ? 2. In a factory, there are three machines 1, 2,
3, producing 50%, 30%, 20% respectively of the total
output. Out of the items produced by machine 2, four percent are
defectives. The corresponding
figure for machine 3 is 6%. The following is known:
If an item is drawn at random from the production line and found
to be defective then the conditional probability for this item to
be produced by machine 1 is 0.50. What is the proportion of
defective items among those produced by machine 1? 3. An
anti-aircraft gun can fire up to a maximum of four shots at an
enemy plane moving away
from it. If the probabilities of hitting the plane at the first,
second, third and fourth shots are 0.4,
0.3, 0.2 and 0.1 respectively, what is the probability that the
gun is able to hit the plane ? Given that
the plane is hit, what is the conditional probability that at
least three shots are needed for this
purpose? 4. Out of the valves produced by factory A, 10% are
defectives and the corresponding figure for
factory B is 20%. A bag contains 4 valves of factory A and 5
valves of factory B. If two valves are
drawn at random from the bag, find the probability that at least
one valve is defective. 5. The following table provides information
on brand preference separately for adult males and
females. For instance, 60% of adult males prefer brand A to
brand B, and so on.
Prefer brand A to brand B Prefer brand B to brand A
Men 60% 40%
Women 90% 10%
Assume that the adult gender ratio in the population is 1:1. A
random sample of n adults is drawn.
(a) What is the probability that all these n people prefer brand
A to brand B?
(b) Now suppose n = 3. Given that all these three people prefer
brand A to brand B, what is the
conditional probability that exactly two of them are males? 6. A
business school conducts an examination on quantitative methods in
two rounds. Each student
must appear in the first round of the examination, and any such
student is assigned one of the four
letter grades A, B, C or F, on the basis of his/her performance
at this round. It is known that 10%,
30%, 40% and 20% of the students get A, B, C and F grades
respectively, at the first round. If a student gets an A or a B
grade at the first round then this is recorded as his/her final
grade.
Otherwise, he/she has the option of sitting for the second round
of the examination for possible grade
improvement. Among the students securing a C grade at the first
round, 40% do not opt for the second round of
examination and for them the final grade is recorded as C. On
the other hand, among those who get a
C grade at the first round and decide to sit for the second
round, 10%, 10%, 50% and 30% get A, B, C
and F grades respectively, at the second round. All those
securing an F grade at the first round sit for the second round of
examination and, among
them, 5%, 10%, 40% and 45% get A, B, C and F grades
respectively, at the second round. As per the
rules of the business school, the grades at the second round are
converted to final grades using a
formula which is summarized as follows:
Grade in the second round A B C F
Final grade B C C F
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(a) What is the probability that a randomly chosen student sits
for the second round of the
examination?
(b) What is the probability that a randomly chosen student gets
a final grade C ?
(c) Given that a randomly chosen student has a final grade B,
what is the conditional probability that
he/she did not sit for the second round of examination?
(d) Given that a randomly chosen student has a final grade F,
what is the conditional probability that
he/she got an F grade in the first round of examination? 7. A
production process involves three machines A, B and C, which
produce 50%, 30% and 20%
respectively, of the total output. Out of the items produced by
machine A, 10% fail in a quality control
test. The corresponding figures for machines B and C are 20% and
30% respectively. All items passing
the quality control test are directly acceptable. On the other
hand, items failing in the quality control
test are further processed and thus 40%, 50% and 60% of them
turn out to be marginally acceptable,
depending on whether they came from machines A, B and C
respectively, e.g., out of the items, that
are produced by machine A and that fail in the quality control
test, 40% eventually turn out to be
marginally acceptable, and so on. (a) Find the probability that
a randomly chosen item from the production process is found to
be
directly acceptable.
(b) Find the probability that a randomly chosen item from the
production process turns out to be
marginally acceptable.
(c) Given that a randomly chosen item from the production
process has failed in the quality control
test, what is the conditional probability that it turns out to
be marginally acceptable?
(d) Given that a randomly chosen item from the production
process has turned out to be marginally
acceptable, what is the conditional probability that it was
produced by machine A?
(e) Given that a randomly chosen item was not produced by
machine B, what is the conditional
probability that it turns out to be marginally acceptable? 8.
The e-mail system of a management institute has a sensor that
evaluates the seriousness of each
incoming e-mail to any account and then places it either in the
inbox or the junk-box of that account.
Consider three accounts A, B and C with this system. Any
incoming e-mail to these accounts is either
educational (called E-type for brevity) or of other kinds
(called O-type for brevity). The following are
known:
The sensor puts any E-type e-mail either in the inbox or in the
junk-box with respective probabilities 0.8 and 0.2.
The sensor puts any O-type e-mail either in the inbox or in the
junk-box with respective probabilities 0.3 and 0.7.
Incoming e-mails are placed in the inbox or junk-box
independently of one another. (a) It is known that 60% of the
e-mails coming to account A are of E-type and the rest are of
O-type.
An e-mail was picked up at random from all the incoming ones to
this account. It was seen that this e-
mail was one of those stored in the junk-box. What is the
conditional probability that this e-mail was
of O-type? (b) It is known that the number of E-type e-mails
that arrive at account B on any day, during the
period 8:00am 9:00am, equals 1, 2 or 3 each with probability
1/3. Obtain the probability that among such e-mails on a particular
day, exactly two go to the inbox. (c) The following information is
available about account C:
Everyday, during the period 8:00am 9:00am, it receives exactly
one e-mail of the E-type.
Furthermore, during the same period, it receives either no
e-mail of the O-type or exactly one e-mail of the O-type, with
respective probabilities 0.4 and 0.6.
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Among the incoming e-mails to this account on a particular day
during 8:00am 9:00am, suppose X go the inbox and Y go to the
junk-box. (I) Calculate P(X = 1). (II) Calculate P(Y = 1| X = 0).
9. The final round of the admission test for a management institute
consists of three components: (I)
group discussion, (II) interview with practicing managers, (III)
interview with the faculty of the
institute. The city, where the test is held, is notorious for
traffic snarls. If a candidate is held up in a
traffic jam on the way to the test venue, he gets mentally
disturbed and that can adversely affect his
performance in the test. In this case, his chances of being
successful in (I), (II) and (III) are 0.3, 0.5
and 0.2 respectively. Otherwise, the corresponding chances are
0.6, 0.8 and 0.5 respectively. The
performances in (I), (II) and (III) in either case can be
supposed to be mutually independent. The
chance that the candidate encounters a traffic jam on the way to
the test venue is 0.3. (a) Given that the candidate did not
encounter a traffic jam on the way to the test venue, what is
his
conditional probability of being successful in at least two of
the three components of the test?
(b) Given that the candidate encountered a traffic jam on the
way to the test venue, what is his
conditional probability of being successful in at least two of
the three components of the test?
(c) Given that the candidate was successful in at least two of
the three components of the test, what is
the conditional probability that he did not encounter a traffic
jam on the way to the test venue? 10. Mr. Travel Nord has a special
interest in the three countries Denmark, Sweden and Norway. In
2012 he visits one of these countries with respective
probabilities 0.4, 0.3 and 0.3. There is a 20%
chance that he develops a strong liking for the country visited
in 2012, in which case he makes a repeat
visit to the same country in 2013. Otherwise, he picks up one of
the two other countries with equal
chance for a visit in 2013. If Mr. Nord visits the same country
in 2012 and 2013, then in 2014 he visits
one of the two other countries with equal chance. Otherwise, in
2014 he simply visits the remaining
country. (a) Find the probability that Mr. Nord visits three
different countries in the three years.
(b) Find the probability that Mr. Nord does not visit Denmark in
2014.
(c) Given that Mr. Nord will visit Sweden in 2014, what is the
conditional probability that he visits
Norway in 2012?
(d) Given that Mr. Nord does not visit Norway in 2012, what is
the conditional probability that he will
visit Denmark in 2014?
(e) Given that Mr. Nord visits exactly two different countries
in the three years, what is the conditional
probability that he does not visit Denmark in 2013? 11.
Professor Notso Bookworm teaches in a leading business school. He
also has a strong
commitment to the society and is involved with several NGOs. On
Sunday, September 14, 2003, he replies to all pending e-mails and
his inbox is empty at the end
of the day. The number, X, of new NGO-related e-mails that he
receives on September 15, 2003,
has the following probability distribution:
X 1 2 3 Total
P(X = x) 0.3 0.4 0.3 1 The professor is scheduled to attend an
academic meeting on September 15, 2003. The probability
for the meeting to be over within two hours is 0.3. In this
case, he sends a reply to any new
incoming e-mail on the same day with probability 0.8. Otherwise,
this probability is only 0.4. It may be assumed that the new
e-mails are treated independently. Thus if X = 2 and the meeting
is
over within two hours then the chance that both the new
NGO-related e-mails remain unanswered
on September 15 is (1 0.8)2. Also, it is perfectly legitimate to
assume that the number of new incoming NGO-related e-mails is
independent of the duration of the meeting.
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(a) Given that the meeting on September 15 was not over within
two hours, what is the conditional
probability that the professor did not reply to any of the new
incoming NGO-related e-mails on that
day?
(b) What is the (unconditional) probability that the professor
received at most two NGO-related
new e-mails on September 15 and did not reply to any of them ?
12. In 2008, there were three brands A, B and C of a product having
market shares 20%, 30% and 50%
respectively among a group of 1000 consumers. A new brand D came
into the market in 2009. The
following are known about the behavior of these 1000
consumers:
(i) Among the consumers of brands A, B and C in 2008,
respectively 50%, 40% and 30% switch over
to brand D in 2009. These people continue to use brand D till
the end of 2009, but eventually some of
them start disliking brand D. As a result, among the converts to
D from A, B and C, respectively 60%,
30% and 40% return to their original brands on January 1,
2010.
(ii) Among the three brands A, B and C, there is no change of
loyalty in 2009 (i.e., there is no
conversion from brand A to brand B, and so on).
(iii) Those who do not change brand loyalty in 2009 continue
with their original brands on January1,
2010.
Direction for parts (a)-(c): On December 31, 2009, one of the
1000 consumers is chosen at random.
(a) What is the probability that the chosen person is a consumer
of brand D?
(b) Given that the chosen person is not a consumer of brand D,
what is the conditional probability that
he/she was a consumer of brand C in 2008 ?
(c) Given that the chosen person is not a consumer of brand C,
what is the conditional probability that
he/she will be a consumer of brand A on January 1, 2010 ?
Direction for parts (d)-(f): On January 1, 2010, one of the 1000
consumers is chosen at random.
(d) What is the probability that the chosen person is a consumer
of brand D?
(e) What is the probability that the chosen person is a consumer
of brand C ?
(f) Given that the chosen person is a consumer of brand A or C,
what is the conditional probability that
he/she did not change brand loyalty in 2009 ? 13. It is known
that 50% of the passengers of an airline are Indians. The rest are
obviously foreigners.
The passengers arrive at the check-in counter at a random order.
Consider the first four passengers
who check-in for the flight. If the first passenger is an Indian
then we write I for him or her; otherwise
we write F. The same thing is done for the second, third and
fourth passengers. This gives a sequence
of length four consisting of I and F. For example, if all the
first four passengers are foreigners then we
get FFFF. On the other hand, if the first and fourth passengers
are Indians and the second and the third
passengers are foreigners, then we get the sequence IFFI. In any
such sequence, an F-run is an
uninterrupted chain of Fs preceded and followed by either I or
nothing. Similarly, an I-run is defined. Let X and Y denote the
numbers of F- and I- runs respectively. Thus with the sequence
IFFI, we have
X=1 and Y=2 since the first and last members of the sequence
give two I-runs whereas the two middle
members yield an F-run. Here are some more examples
corresponding to several other possibilities for
the sequence formed by the first four passengers:
Sequence X Y
FFFF 1 0
IIFF 1 1
FIFI 2 2 Obtain (a) P(X=0 | Y=1), (b) P(X=1 | Y=1) and (c) P(X=2
| Y=1). 14. The distribution of life (in hours) of a certain kind
of electric bulb is known to be exponential. It is
known that any bulb of this kind survives for 100 hours or more
with probability 0.6561. What is the
probability for any bulb of this kind to fail within 75
hours?
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15. (a) The waiting time (in minutes) at a bus stop follows the
uniform distribution over the range
[0,10]. What is the probability that the total waiting time,
over 30 occasions, exceeds 150 minutes?
(b) The distribution of a certain quality characteristic is
continuous uniform over [ 8, 4]. If M is the arithmetic mean of
quality characteristic for 48 randomly chosen items, then find )94(
MP .
(c) The distribution of scores in a public examination is known
to be normal with mean and standard
deviation . (i) If 20% of the candidates score over 80 and 30%
of the candidates score below 40, then find the ratio / . (ii) If
44 and 7 , then find the probability that among 100 randomly
chosen candidates no more than one scores below 29. 16. The
probability distribution of the number of projects executed by a
construction company per year
is as follows:
Value 4 5 6 7 8 9 10
Probability k2 k 7k
2 +k 2k 2k
2 3k 2k
Here k is a suitable constant. Any year is "unusual" if the
number of projects executed, X, is too large
or too small in the sense that |X7| > 2. (a) Find the
probability for any particular year to be "unusual".
(b) Also find the probability that out of four randomly chosen
years, no more than one is "unusual".
(c) Find E(X). 17. There are three varieties of an insect. These
varieties are called A, B and C and they occur in equal
proportion in the nature. An entomologist is conducting research
on this insect and is primarily
interested in variety A. He collects the insect one-by-one till
an insect of variety A is obtained. The
process, however, is not allowed to continue indefinitely. If an
insect of variety A is not obtained even
in four trials, the entomologist stops collecting any more
insect. Find the expected number of insects
collected. 18. There are two machines A and B in a factory. The
number of defectives, say X, produced by
machine A on a particular day follows the Poisson distribution
with mean 2. Also, the number of
defectives, say Y, produced by machine B on the same day equals
0, 1, 2 or 3, each with probability
0.25. Independence of the two machines can be assumed.
(a) Given that exactly two defective items were produced on that
day, find the conditional probability
that one of these defective items came from machine A and the
other from machine B.
(b) Given that at least one of the machines produced two or more
defective items on that day, what is
the conditional probability that one of the two machines did not
produce any defective item at all on
that day? 19. The profit (in a certain unit) of a business
enterprise for a particular year equals the larger root of
the quadratic equation 02 BAxx , where A and B are two economic
indicators, about which the following are known:
A equals either 7 or 8 with respective probabilities 0.4 and
0.6.
Given A = 7, the possible values of B are 10 or 12 with
respective conditional probabilities 0.7 and 0.3.
Given A = 8, the possible values of B are 12 or 15 with
respective conditional probabilities 0.8 and 0.2.
Let Z be the profit for the year.
(a) Obtain the conditional probability P(B = 15 | Z = 5). (b)
Calculate E(Z). 20. The length of life (in years) of a certain type
of automatic washer is normally distributed with a
mean of 3.1 and a standard deviation of 1.2. If this type of
washer is guaranteed for two years, what
fraction of original sales will require replacement? Also,
calculate the probability that out of a total of
240 such machines, at least 115 will require replacements as a
consequence of the guarantee.
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21. A factory manufactures electric bulbs of two brands A and B.
The distribution of life (in hours) of
any bulb of brand A is uniform over the range (0,4). The
corresponding distribution for brand B is
uniform over the range (1,5). Four bulbs, of which two are of
brand A and two are of brand B, are put
to test in a life testing experiment. Let X be the number of
bulbs that fail during the first two hours of
the experiment and Y be the number of bulbs that fail during the
first three hours of the experiment.
Find E(Y X) and P(Y=3). 22. A zoologist, conducting research on
crocodiles, collects crocodiles in a mangrove forest. There are
two kinds of crocodiles in the forest, interesting and boring
(depending on a certain characteristic), in
the ratio 1:2. The crocodiles are collected one by one and at
random. The population of crocodiles in
the forest is large so that the successive trials may be assumed
to be independent.
(a) What is the probability that the first interesting crocodile
is obtained at the fifth trial?
(b) What is the probability that at least six trials are needed
to get the fourth boring crocodile?
(c) What is the expected number of interesting crocodiles that
are collected before the third boring
crocodile is obtained? (no derivation needed; any relevant
formula may be stated without proof).
(d) Suppose the third interesting crocodile appears in the Xth
trial and the fifth interesting crocodile
appears in the Zth trial. Find E(Z X). (no derivation needed;
any relevant formula may be stated without proof). 23. The
distribution of life (X), in hours, of a certain kind of electrical
component has the
probability density function
)()( axkxf , if 2 axa ,
= 0, otherwise,
where k and a are constants. It is known that E(X) = 2. Obtain
the values of k and a. Also find the
median of X, the variance of X and P(X > 1a |23
21 aXa )
24. The number of defects on a particular brand of plastic sheet
(of fixed length and breadth) follows
the Poisson distribution. Out of four such sheets chosen at
random, let
X = number of sheets with no defects,
Y = number of sheets with exactly one defect,
Z = number of sheets with exactly two defects.
It is known that the expectation of Z is twice that of X.
Find
(a) the variance of the number of defects on the second of the
four randomly chosen sheets,
(b) the variance of the number of sheets (among the chosen four)
with two or more defects,
(c) P(X=1|Y=3). 25. Let X and Y denote the numbers of defective
spots on two plastic sheets. The distribution of X is
Poisson such that P(X = 2) = P(X = 1). The distribution of Y is
uniform with possible values 1, 2,, n, such that E(Y) = 3. Assume
that X and Y are independent.
(a) Find the probability that two sheets together have three or
more defective spots.
(b) Given that the two sheets together have exactly five
defective spots, what is the conditional
probability that none of them has more than three defective
spots?
(c) In a study on quality improvement, interest lies in the
quantity Z, defined as 4/)( YXZ for
3Y and 4/XZ for 3Y . Find E(Z). 26. On the basis of an
assessment of the market condition for the next financial year, it
is envisaged
that the scenario can be of four possible types, coded 0,1,2 and
3, where 0 represents the worst
scenario and 3 stands for the best. Of course, these four
possibilities are mutually exclusive and
exhaustive. It is also estimated that the worst and best
scenarios have probabilities 0.2 and 0.1
respectively. The profit of a company, as a function of the
investment x, under the scenario coded s , is
)(xsP = 5 + xs3)1( + 2)2( xs ( s 0,1,2,3)
If the expected profit is maximum at x 1, then what is the
probability that scenario 2 will arise?
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27. A travel agent in a hilly region offers helicopter rides.
The agent has only one helicopter which
can accommodate up to 4 passengers on any one trip. The agent
will accept a maximum of 8
reservations for a trip and a passenger must have a reservation.
From previous records, 50% of all
those making reservations do not appear for their trip.
(a) If six reservations are made for a trip, what is the
expected number of empty seats when the
helicopter departs ?
(b) Suppose that the expected number of reservations made for
any trip is 6, and that the probability
mass function of the number of reservations made on a randomly
selected trip is given by
f(x)=c+dx, x=5,6,7,8 and f(x) = 0 otherwise. Let Y denote the
number of passengers on a randomly
selected trip. Obtain the values of c and d. Also find P(Y = y),
for y= 0, 1, 2, 3 and 4. 28. A system consists of five identical
components connected in series so that the entire system fails
as
soon as any one of the components fails. The distribution of
life (in hours) of each component is
exponential with median 90. Let X be the life of the system and
Y be the arithmetic mean of lives of
36 such systems. Obtain E(X), V(X) and P(Y > 30). 29. A
factory produces two types of electric bulbs A and B. Sixty percent
of the total output are of
type A and the rest are of type B. Any bulb of type A survives
for 40 hours or more with probability
0.64. The corresponding probability for any bulb of type B is
0.36. It is known that the life distribution
of any bulb, either of type A or of type B, is exponential.
(i) Consider two bulbs, one of type A and the other of type B.
What is the probability that they both
survive for 20 hours or more?
(ii) Given that a randomly chosen bulb from the total output of
the factory has failed within 60 hours,
what is the conditional probability that it is of type A? 30.
The following information is available about the graduating batches
of a business school during the
period 2004-06.
Year 2004 2005 2006
Percentage of students securing a job
with a multinational
40 20 30
One student is chosen at random from the graduating batch of
each year. Let X be the number of
students, out of the three so selected, who had secured a job
with a multinational. Obtain P(X = x) for
x = 0, 1, 2, 3. Also find E(X) and V(X). 31. The proportions of
tall people in four regions of a country, say North, East, South
and West, are
0.5, 0.3, 0.3 and 0.4 respectively. One person was chosen at
random from each region. Assume
independence across regions. Let
X = number of tall persons among the four people so
selected,
Y = number of tall persons among the two people selected from
East and South.
Calculate (a) V(X), (b) P(X = 2), (c) )( 2YE , and (d) V(Y | X =
3). 32. The number of projects (X) that a company bids for per year
equals 3, 4 or 5, with respective
probabilities 0.3, 0.4 and 0.3. . If a bid is made for any
project, the company gets it with probability
0.5. Independence across different projects can be assumed
safely. Let Y be the number of projects
that the company gets in that year. Obtain (a) P(Y = 4), (b) P(X
= 5 and Y 2), (c) P(X = 5| Y = 3)
and (d) E(X Y). 33. The probability distribution of the number
of e-mails, X, received daily by a businessman is as
follows:
P(X = x) =
4
11
4
3x
, x =1, 2, 3,
On a particular day, if two or fewer e-mails are received then
the businessman replies to these e-mails
immediately. However, if three or more e-mails are received then
only the first three of these are
replied to on the same day. Define
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Y = number of e-mails, among those received on that day, that
are replied to on the same day.
Z = number of e-mails, among those received on that day, that
are not replied to on the same day.
Obtain (a) P(Y = 2), (b) P(Y = 3), (c) P(Z = 1 | Y =3), (d)
E(Y), (e) Var(Y), and (f) E(Z). 34. Let X denote the number of
projects that a company bids for in a particular year and Y denote
the
number of projects, out of these X, that the company gets. It is
known that the possible values of X are
2, 3 or 4 with respective probabilities 0.4, 0.4 and 0.2.
Furthermore, for x = 2, 3 or 4, given X = x,
conditionally Y equals either x or x-1 each with probability
0.5. Obtain (a) P(X = 2 | Y = 2), (b) P(X =
3 | Y = 3), (c) E(X | Y =2), (d) Var(X | Y =2), (e) E(Y) and (f)
E(XY). 35. The market condition in a region can be upbeat, moderate
or poor, with respective probabilities
0.3, 0.5 and 0.2. The following are known about the profit, say
X (in a certain monetary unit), of a
company under various market conditions:
(i) Upbeat: X equals 2, 3 or 4 with respective probabilities
0.2, 0.3 and 0.5;
(ii) Moderate: X equals 1, 2 or 4 with respective probabilities
0.3, 0.5 and 0.2;
(iii) Poor: X equals 1, 2 or 3 with respective probabilities
0.5, 0.3 and 0.2.
If X equals 3 or 4, then the company may be fined by the
regulatory authorities for restrictive trade
practices, the amount of fine being 1 monetary unit. If 3X ,
then this happens with probability 0.3, in which case the net
profit, say Y (in the same monetary unit as X), equals 2. On the
other hand, if
4X , then the fine is imposed with probability 0.6, in which
case the Y equals 3. Of course, with X = 1 or 2, the question of
imposition of a fine does not arise and hence Y equals X. The same
happens
if X equals 3 or 4 and no fine is imposed.
(a) Find the probability that the company has to pay a fine.
(b) Obtain the conditional probability ).2|3( YXP
(c) Given that the market condition was moderate, what is the
conditional expectation of Y ?
(d) Find the conditional variance of X given 3Y . 36. A factory,
making a certain electrical component, has two machines, each
producing 50% of the
total output. Any component produced by the first machine
survives for at least the first 40 hours with
probability 0.64. The corresponding probability for any
component produced by the second machine is
0.49. For each machine, the life distribution of components
produced by it is exponential. Two
components are chosen at random from the production line.
(a) What is the probability both of these two components will
fail within the first 60 hours?
(b) Given that both of these components had failed within the
first 60 hours, what is the conditional
probability that they were produced by different machines? 37.
It is known that the life (in hours) of a certain kind of electric
bulb is specified by the following
probability density function:
)(xf = )32(72
1 2x , 0 < x < 4,
Let 1
X ,, 36X be the lives of 36 bulbs of this kind. Define
Z = max (1
X ,2
X ), W = min (1
X , 2X ), Y = 1X + 2X ++ 36X .
Assuming independence across bulbs, obtain (a) P(2 104). 38. A
psychological trait, sat X, is known to influence consumer
behavior. Persons with 66 X
have normal behavior, while those with 6X are hyperactive and
those with 6X are hypoactive.
Parts (a) (e) below concern different geographic locations. (a)
In location A, the distribution of X is specified by the
probability density function
)(xf||
2
xe
, x ,
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9
where is a positive constant. If 80% of the people in this
location have a normal behavior, then what is the value of ? (b) In
location B, the distribution of X is specified by the probability
density function
)(xf ))(()(
63
xbaxab
, if bxa ,
= 0, otherwise,
where a and b are positive constants.
(i) For this location, if 21)( XE and 7a then what is the
minimum possible value of b ?
(ii) If instead 10a and 14b , then what is the probability that
among four randomly chosen people from this location, at least
three will have the normal behavior? (c) In location C, 10% of the
people are hyperactive and 20% of the people are hypoactive.
Furthermore, the proportions of buyers of a particular brand
among the hyperactive, normal and
hypoactive people are 0.8, p and 0.05 respectively. The
following is also known about location C:
Given that a randomly chosen person is a buyer of this
particular brand, the conditional probability of his/her being
hyperactive is 0.4. (i) What is the value of p ?
(ii) Will the answer in (i) change if the distribution of X in
location C is known to be normal? (d) In location D, the
distribution of X is unknown but it is known that 0)( XE and 4)(
XVar .
From this information alone, the marketing manager of a company
concludes that less than 12% of the
people in this location are hyperactive. Is the manager correct?
(e) In location E, the distribution of X is normal. It is also
known that in this location, 10% of the
people are hyperactive and 20% of the people are hypoactive.
What are the values of the mean and
standard deviation of X for this location? Alternatively, do you
think that it is impossible to find any
of these from the given information alone? 39. A study on a
psychological trait, X, affecting consumer behavior is conducted
over three cities
Jaipur, Patna and Kochi. In Jaipur, X is exponentially
distributed with mean 3, while in Patna, the
distribution of X is continuous uniform over the range ],4[ h ,
where h is a positive constant. Finally,
the distribution of X in Kochi is specified by the probability
density function ||)( xkxf if 4|| x ,
and 0)( xf otherwise, where k is a positive constant.
(a) A person is called hyperactive if his/her X-value exceeds
11. Find the probability that out of 100
randomly chosen people in Jaipur at least two are
hyperactive.
(b) If V(X) = 3 in Patna, then find P(X > 0 | 3 < X <
3) for this city. (c) Find )13( XP for Kochi. 40. (a) The number of
patients, say X, arriving at a specialty clinic on a particular day
follows the
Poisson distribution with mean two. The clinic can handle at
most two such patients. Thus if Y denotes
the number of patients handled by the clinic, then Y = X if X 2,
while Y = 2 if X > 2. Find the expectation of Y. (b) Let X be
the number of defectives among four items drawn at random from a
large lot. The lot is
accepted if X = 0, and rejected if X 2. If X = 1, then two more
items are drawn at random from the lot. The lot is rejected if both
these items are defectives, and accepted otherwise. If 20% of the
items in
the lot are defectives, then what is the probability of
acceptance of the lot? (c) A system, with three components A, B and
C arranged in series, functions if and only if all the
three components function. The life (in hours) of each component
has the same exponential
distribution and these components behave independently. The
probability that the system functions for
at least 8 hours is 0.6561. What is the probability that
component A functions for 6 hours or more?
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10
(d) In an examination, where the distribution of scores is
normal, the proportion of candidates getting
70 or more exceeds the proportion of candidates getting 30 or
less. Which of the following best
describes the mean of the distribution?
A. 50 B. 5030 C. 50 D. 7050 (e) The demand, X, of a certain
commodity is known to have the continuous uniform distribution
over
the range [8, 12]. The profit, Y, is related to X as follows: Y
= X+1 if X 10, while Y = X2 otherwise. What is the expectation of Y
?
41. Let X denote the life (in hours) of a certain kind of bulb
which is produced in four locations A, B,
C and D. The following are known:
(i) At location A, the distribution of X is exponential with
mean 10.
(ii) At location B, the distribution of X is continuous uniform
with mean 10 and variance 12.
(iii) At location C, the distribution of X is specified by the
probability density function
)(xf = 2)12( xk , if 128 x ,
= 0 otherwise,
where k (> 0) is a constant.
(iv) Ten per cent of the bulbs produced at location D are
defectives (i.e., fail within 5 hours). (a) If two bulbs are
selected at random from the output of location A, then what is the
probability that
one will survive for 10 hours or more and the other will fail to
do so? (b) If one bulb is selected at random from the output of
location B, then what is the probability that it
will survive for 12 hours or more? (c) With reference to
location C, what is the value of k ? (d) The manufacturing cost at
location D is Rs 10 per bulb and selling price of any bulb
produced
here is Rs 12. However, if any bulb sold turns out to be
defective then a complete refund is made to
the customer. What is the variance of the profit (in Rs) for any
bulb sold from the output of location
D ? (e) From the output of location D, bulbs are inspected one
by one at random till a defective bulb is
found. Let Y be the number of non-defective bulbs inspected in
the process. A performance measure
Z is defined as Z = 2 if Y > 1, and Z = Y otherwise. What is
the value of E(Z) ? 42. The joint probability distribution of two
business indicators X and Y, each with three possible
values 1, 2 and 3, is shown below:
P(X = 1 and Y = 1) = 0.1, P(X = 1 and Y = 2) = 0, P(X = 1 and Y
= 3) = 0.2,
P(X = 2 and Y = 1) = 0, P(X = 2 and Y = 2) = 0.3, P(X = 2 and Y
= 3) = 0.1,
P(X = 3 and Y = 1) = 0.2, P(X = 3 and Y = 2) = 0.1, P(X = 3 and
Y = 3) = 0.
Find (a) E(Y), (b) E(XY), (c) E(X| Y = 2) and (d) Var (Y| X= 3).
43. Consider the probability distributions of a quality
characteristic X at three different locations of a
manufacturing unit.
(a) At location 1, the distribution of X is continuous uniform
over the range [ 2, 3]. Three items are drawn at random from the
production line at this location, and let Y be the minimum of the
X-values
for these three items. Find the conditional probability P(Y <
0| Y < 2).
(b) At location 2, if X is exponentially distributed and P(X
> 30) = 0.216, then find P(X < 40).
(c) At location 3, the distribution of X is specified by the
probability density function )(xf , where
)(xf = |2| xk if 41 x , and )(xf = 0 otherwise. Find E(X) at
this location.
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11
STATISTICS FOR MANAGEMENT
PROBLEMS ON PROBABILITY AND DISTRIBUTIONS
ANSWERS
1. 7/9
2. 0.048
3. 0.6976, 0.1686
4. 0.2872
5. (a) (0.75)n, (b) 0.288
6. (a) 0.44, (b) 0.404, (c) 0.8982, (d) 0.5556
7. (a) 0.83, (b) 0.086, (c) 0.506, (d) 0.232, (e) 0.08
8. (a) 0.7, (b) 0.3413, (c) (I) 0.692, (II) 0.4878
9. (a) 0.7, (b) 0.25, (c) 0.8673
10. (a) 0.8, (b) 0.7, (c) 0.4286, (d) 0.2143, (e) 0.6
11. (a) 0.3888, (b) 0.2496
12. (a) 0.37, (b) 0.5556, (c) 0.246, (d) 0.214, (e) 0.410, (f)
0.789
13. (a) 0.1, (b) 0.6, (c) 0.3
14. 0.271
15. (a) 0.5, (b) 0.3085, (c) (i) 1.8904, (ii) 0.5217
16. (a) 0.21, (b) 0.8037, (c) 7.82
17. 2.407
18. (a) 0.4, (b) 0.2712
19. (a) 0.3, (b) 5.36
20. 0.1798, 0
21. E(Y X) = 1, P(Y=3) = 0.375 22. (a) 0.0658, (b) 0.5391, (c)
1.5, (d) 6
23. k = 0.5, a = 2/3, median = (2/3)+ 2 , V(X) = 2/9,
conditional probability = 5/8
24. (a) 2, (b) 0.9647, (c) 0.1856
25. (a) 0.8917, (b) 0.4762, (c) 0.8
26. 0.4667
27. (a) 1.125, (b) c = 0.9, d = 0.1, P(Y=0)= 0.0191, P(Y=1)=
0.1047, P(Y=2)=0.2391, P(Y=3)= 0.2953, P(Y=4) = 0.3418
28. E(X) = 25.9685, V(X) = 674.3635, P(Y > 30) = 0.176
29. (i) 0.48, (ii) 0.4828
30. P(X=0)= 0.336, P(X=1)= 0.452, P(X=2)= 0.188, P(X=3)=
0.024,
E(X)= 0.9, V(X) = 0.61
31. (a) 0.91, (b) 0.335, (c) 0.78, (d) 0.2271
32. (a) 23/320, (b) 0.15, (c) 15/37, (d) 2
33. (a) 3/16, (b) 9/16, (c) 3/16, (d) 37/16, (e) 0.7148, (f)
27/16
34. (a) 0.5, (b) 2/3, (c) 2.5, (d) 0.25, (e) 2.3, (f) 7
35. (a) 0.189, (b) 0.095, (c) 1.98, (d) 0.235
36. (a) 0.3278, (b) 0.4891
37. (a) 7/24, (b) 26/9, (c) 77/192, (d) 49/576, (e) 1/2
38. (a) 0.2682, (b) (i) 8, (ii) 16/27, (c) (i) 0.1571, (ii) No,
(d) Yes, by Chebycheffs inequality, (e) mean = 1.2427, standard
deviation = 5.6513 39. (a) 0.7240, (b) 0.4, (c) 0.3125
40. (a) 1.4586, (b) 0.8028, (c) 0.9, (d) correct alternative is
C, (e) 65.667
41. (a) 0.4651, (b) 1/3, (c) 3/64, (d) 12.96, (e) 1.71
42. (a) 2, (b) 3.7, (c) 2.25, (d) 2/9 43. (a) 0.7903, (b)
0.8704, (c) 2.9333
