Question Related to probability, Random variables, Binomial
distribution, hyper geometric distribution, and Normal distribution
Q#1(a): Find the number of ways in which all twelve letters of the word “REFRIGERATOR”
can be arranged
(i) If there are no restrictions
(ii) If the all R’s must be together.
(b):
How many different selection of four letters from the twelve letters of the word
“REFRIGERATOR” contain no R’s and two E’s?
Q#2:
A vegetable basket contains 12 peppers, of which 3 are red, 4 are green and 5 are
yellow. Three peppers are taken at random from the basket:
(i): Find the probability that all three peppers are of different colours.
(ii): Show that the probability that exactly 2 of the peppers taken are green is
.
(iii): The number of green peppers taken is denoted by the discrete random variable X.
Find the probability distribution for X, also calculate mean and variance the
probability distribution of X.
Q#3(a): Let
, also given that P(X>34) = 0.0228 and P(X<25) =0.0062, then
find the value of
(b):
and
.
A traffic study at one point on highway shows that vehicle speeds (in km/hrs) are
normally distributed with mean 61 and standard deviation 4. If a vehicle randomly
checked, what is the probability that its speed is:
(i): Below 55 km/hr.
(ii): Between 53 and 59 Km/hr
(iii): Find the two speed limits between which 70% vehicles having their speeds.
(iv): Find a point below which 82% vehicles having their speeds.
Q#4(a): A pheasant hunter brings down 68% of the birds he shoots at. What is the probability
that at least 3 of the next 5 pheasants shoot at will escape?
(b):
What is the mean and variance of the number of Pheasant he shoots at will not
escape?
Q#5:
A random variable x has the p.d.f as
(i)
Find the value of A
(ii)
After replacing the calculated value of A find mean and standard deviation for the
p.d.f.
Q#6: A manufacturer of Cotter pins knows that 2 percent of his product is defective. If he sells
cotter pins in boxes of 100, and guarantees that not more than 4 pins will be defective,
what is the probability that box will fail to meet the guaranteed quality?
Q#7: The following probability distributions of job satisfaction scores for a sample of
information system (IS) senior executives and IS middle managers range from a low of 1
(very dissatisfied ) to a high of 5 (very satisfied).
Job satisfaction score
IS Senior Executives
IS Middle Managers
1
0.05
0.04
2
0.09
0.10
3
0.03
0.12
4
0.42
0.46
5
0.41
0.28
(a): What are the expected values of the job satisfaction score for senior executives and
middle managers?
(b): Compare the consistency for the senior executives and middle managers, according to
your opinion which group is more consistent and why?
Q#8
The phone lines to an airline reservation system are occupied 40 percent of the time.
Assume that the events that the lines are occupied on successive calls are
independent. If ten calls are placed to the airline then:
(i): What is the probability that for exactly three calls the lines are occupied?
(ii): What is the probability that for at least two calls the lines are not occupied?
(iii): What is the expected number of calls in which the lines are occupied?
Q#9: If in a normal distribution 31% of the items are fewer than 45 and 8% are more than 64.
Find its mean and variance.
Q#10: In a city of Multan 35% households have a freezer and 60% have colour T.V. set. Given
that 25% of the households have both freezer and colour T.V set, find the probability that
a household has either a freezer or a colour T.V. set but not both.
Q#11: A bin of 50 manufactured parts containing 3 defectives and 47 non defective parts. A
sample of six parts is selected from the 50 parts. Selected parts are not replaced. That is,
each part can only be selected once and the sample is a subset of 50 parts. How many
different samples are of size six that contain exactly two defectives parts?
Q#12: The probability that the first stage of numerically controlled machining operation for
high-rpm pistons meets specifications is 0.90. Failures are due to metal variations, fixture
alignment, cutting blade condition, vibration and ambient environment condition. Given
that the first stage meets the specification, the probability that the second stage machining
meets specification is 0.95. What is the probability that both stages meet the
specification?
Q#13:
If on the average rain falls on twelve days in every thirty, find the probability that
(i): the first three days of a given week will be fine and remaining wet.
(ii): Rain will fall on just three days of a given week.
Q#14:
A random variable X follows binomial distribution with n=5,
such that P(X=2) =3P(X=3), then Calculate P(X >2).
Q#15: Suppose a special type of small data processing firm is so specialized that some have
difficulty making a profit in their first year operation. The pdf that characterizes
proportion Y that makes the profit is given by:
f(y)= Ky(1-y)2 0< y < 1
(a): What is the value of “K” that renders the above a valid density function?
(b): Find the probability that at most 40% of the firms make a profit in the first year.
Q#16(a): An industrial plant is conducting a study to determine how quickly injured workers are
back on the job following injury. Records show that 15% of all injured workers are
admitted to the hospital for treatment and 20% are back on the job the next day. In
addition, studies show that 3% are both admitted for hospital treatment and back on the
job the next day. If a worker is injured, what is the probability that the worker will either
be admitted to the hospital or back on the job the next day or both?
(b): A part is labeled by printing with four thick lines, three medium lines, and two thin lines. If
each ordering of the nine lines represents a different label, how many different labels can
be generated by using this scheme?
Q#17: Given P(A) = 0.5 and P(AUB) = 0.6, Find P(B) if
(i): A and B are mutually exclusive.
(ii): A and B are independent.
(iii): P(A/B) = 0.4
Q18: In a certain assembly plant 3 machines M1, M2, and M3, make 30%, 45% and 25%,
respectively of the products. It is known from the past experience that 2%, 3% and 2%, of
the products made by the each machine respectively are defective. A product is selected
at random and found to be defective. What is the probability that the product came from
the machine M3?
Q#19: A box contains 10 red and 12 white rose flowers. Flowers are picked up at random one by
one without replacement. What is the probability that
(i): First three flowers are red?
(ii): There are 2 red and 2 white flowers in the first fours picked up?
(iii): The third one is red flower given that the first 2 are white flowers?
Q#20: A box contains 10 red and 12 white rose flowers. Flowers are picked up at random one
by one without replacement. What is the probability that
(i): First three flowers are red?
(ii): There are 2 red and 2 white flowers in the first fours picked up?
(iii): The third one is red flower given that the first 2 are white flowers?
Q#21: A sample of 500 respondents was selected in a large metropolitan area to study consumer
behavior with the following results:
Enjoys shopping for
cloths
Yes
No
(a):
Gender
Male
136
104
Female
224
36
Suppose the respondent chosen is a female then what is the probability that she does not
enjoy shopping for clothing?
(b):
Q#22:
Suppose the respondent chosen enjoys shopping for clothing then what is the probability
that the individual is male?
Out of 12 eggs in refrigerator, 2 are bad. From these 4 eggs are chosen at random to
make a cake. What is the probability that:
(a): Exactly one egg is bad.
(b): At least one is good egg.
(c): All chosen eggs are bad.
Q#23: In a bolt factory, machines A,B, and C Manufactures 25, 35, and 40 percent of the total
output respectively, of their outputs, 5, 4 and 2 percent respectively are defective bolts. A
bolt is selected at random what is the probability that the bolt is defective?
Q#24: A bookshelf contains 5 German books, 7 Spanish Books and 8 French books. Each book
is distinguishable from one another.
(a): How many different arrangements can be done of these book?
(b):
How many different arrangements can be done of these books if books of each
language must be next to each other?
(c):
How many different arrangements can be done of these books if all the French
books must be next to each other?
Q#25(a): The queuing time, T minutes, for a person queuing at a super market checkout has
probability density function given by,
Where R is a constant
(i):
Show that the value of R is
.
(ii): Find the probability that a person will have to queue between 2 and 4 minutes.
(iii): Find the probability that a person will have to in a queue more than 3.2 minutes.
(b):
The two independent random variables X and Y such that X has mean 8 and variance 4.8,
while Y has a Poisson distribution with mean 6. Find:
(i):
(ii):
Q#26: Find how many different arrangements there are of the nine letters in the words GOLD
MEDAL
(i):
If there is no restriction on the order of the letters.
(ii):
If the two letters “D” come first and the two letters “L” come last.
Q#26:
A husband and his wife appear in an interview for two vacancies of the same post. The
probability of husband’s selection is
and that of his wife’s selection is
. What is
the probability that:
(b):
(i):
Both will be selected
(ii):
At least one of them will be selected.
Given P(A) = 0.60, P(B) = 0.40, P(A∩B) = 0.34. Find the probability that:
(i):
(ii):
P(B/A) =
P(AUB) =
(iii):
What is the relationship between A and B?
Q#27(a): A binomial distribution has mean=1.44 and variance 0.9216. Find P(X > 2)
(b):
A state lottery is conducted in which six winning numbers are selected from a
total 54 numbers. What is the probability that if six numbers are randomly
selected:
(i):
(ii):
Q#28(a):
Four will be winning numbers.
At most one will be winning number.
A machine dispenses liquid into bottles in such a way that the amount of liquid
dispensed on each occasion is normally distributed with standard deviation 20ml. and
mean 266ml. Bottles that less than 260 ml have to be recycled. What percentage of
bottles should be recycled?
(b):
The 10th and 90th percentiles of a certain normal distribution are 17.2 and 42.8
respectively. Find value of Q3, median and standard deviation.
Q#29: A box contains 10 pens of which 3 are new. A random sample of two pens is taken.
(i):
Show that the probability of getting exactly one new pen in the sample is
.
(ii): Construct the probability distribution table for the number of new pens in the sample.
(iii): Calculate the expected number of new pens in the sample.
Q#30(a): The height of sunflowers follows a normal distribution with mean 112cm and
standard deviation 17.2cm. Find the probability that the height of randomly chosen
sunflower is greater that 120 cm.
(b): When a new fertilizer is used the height of sunflowers follows a normal distribution with
mean 115cm. given that 80% of the heights are now greater than 103cm, find the standard
deviation.
(c): Find the point above which 58% heights of the sunflowers heights lie.
Q#31: The queuing time, T minutes for a person queuing at a supermarket checkout has
probability density function given by
Where C is a constant.
(i): Find the value of constant C
(ii): Find the probability that a person will have to queue for between 2 and 4 minutes.
(iii): Find the probability that a person will have to queue more than 3 minutes.
Q#32: Boxes of sweets contains Toffees and Chocolates. Box A contains 6 Toffees and 4
Chocolates, Box B contains 5 Toffees and 3 Chocolates, and Box C contains 3 Toffees and 7
Chocolates. One of the boxes is selected and two sweets are taken out.
(i): Find the probability that they are both toffees.
(ii) If the both sweets taken out are toffees then what is the probability that they both came from
the Box C.
Q#33: The discrete random variable X has the following probability distribution
x
1
3
5
7
P(X = x)
0.3
A
B
0.25
Where “A” and “B” are two constants
(i): Write down an equation satisfied by A and B.
(ii): Given that E(X) = 4, then find the values of A and B
Q#34(a): The height of sunflowers follows a normal distribution with mean 112cm and
standard deviation 17.2cm. Find the probability that the height of randomly chosen sunflower is
greater that 120 cm.
(b): When a new fertilizer is used the height of sunflowers follows a normal distribution with
mean 115cm. given that 80% of the heights are now greater than 103cm, find the standard
deviation.
(c): Find the point above which 58% heights of the sunflowers heights lie.
Q#35: Data about the employment for males and females in a small rural area are shown in the
following table.
Unemployed
Employed
Male
206
412
Female
358
305
A person from this area is chosen at random. Let M be the event that a person is male and let E
be the event that the person is employed.
(i): Find P(M)
(ii): Find P(M and E)
(iii): Are M and E independent? Justify your answer.
(iv): Find the probability that the person is female or someone who is unemployed
Q#36:
From the past experience a stockbroker believes that under present economic conditions
a customer will invest in tax-free bounds with a probability of 0.60, will invest in mutual
funds with a probability of 0.30 and will invest in both tax-free bounds and mutual funds
with a probability of 0.15. Find that probability that the customer will invest in either
tax-free bounds or mutual funds?
Q#37: Three horses A, B, C are in a race. A is twice as likely to win as B, and B is twice as
likely to win as C. What is the probability that A or B wins.
Q#38: An industrial plant is conducting a study to determine how quickly injured workers are
back on the job following injury. Records show that 15% of all injured workers are
admitted to the hospital for treatment and 20% are back on the job the next day. In
addition, studies show that 3% are both admitted for hospital treatment and back on the
job the next day. If a worker is injured, what is the probability that the worker will either
be admitted to the hospital or back on the job the next day or both?
Q#39: An income tax officer has received 10 files numbered from 1 to 10. He selects just one
file for inspection. Find the probability that
(a): the file number is multiple of 5
(b): the file number is multiple of 3
(c): the file number is multiple of 3 or multiple of 5