Download *********************UNIT-1 FAQ********************* 1. If A and B are

*********************UNIT-1 FAQ*********************
1. If A and B are any two arbitrary events of the sample space then prove that P(AUB)= P(A) + P(B) P(AпB).
OR
State and prove addition theorem of probability.
2. Determine the probability for each of the following events:
i. A non defective bolt will be found if out of 600 bolts already examined 12 were defective.
3. Two digits are selected at random from the digits 1 through 9.
ii. If the sum is odd, what is the probability that 2 is one of the numbers selected?
iii. If 2 is one of the digits selected, what is the probability that the sum is odd?
4. Two marbles are drawn in succession from a box containing 10 red, 30 white, 20 blue and 15
orange marbles, with replacement being made after each drawing. Find the probability that
i. Both are white.
ii. First is red and second is white.
5. A businessman goes to hotels X , Y, Z 20%, 50%, 30% of the time resp. It is known that 5%, 4%,
8% of the rooms in X,Y, Z hotels have faulty plumbing. What is the probability that
businessman’s room having faulty plumbing is assigned to hotel Z?
6. Define a random experiment, sample space, event and mutually exclusive events. Give examples
of each.
7. Box A contains 5 red and 3 white marbles and box B contains 2 red and 6 white marbles. If a
marble is drawn from each box, what is the probability that they are both of the same color?
*********************UNIT 1*********************
1. If A and B are events with P(A)=1/3, P(B)=1/4, and P(AUB)=1/2, find
i.
P(A/B)
ii.
P(A
)
2. Three students A,B,C are in a running race. A and B have the same probability of winning and
each is twice as likely to win as C. find the probability that B or C wins.
3. The students in a class are selected at random one after the other for an examination. Find the
probability that the boys and girls are alternate if there are
i.
ii.
5 boys and 4 girls
4 boys and 4 girls
4. If A and B are independent, prove that
i.
A and are independent
ii.
and
are independent.
5. Two bolts are drawn from a box containing 4 good and 6 bad bolts. Find the probability that the
second bolt is good if the first one is found to be bad.
6. State and prove Baye’s theorem.
7. In a certain college 25% of boys and 10% of girls are studying mathematics. The girls constitute
60% of the students. If a student is selected at random and is found to be studying
mathematics, find the probability that the student is a
i. Girl
ii. Boy.
8. (a) Two aero planes bomb a target in succession. The probability of each correctly scoring a hit is
0.3 and 0.2 resp. the second will bomb only if the first misses the target. Find the probability
that
i.
Target is hit
ii.
Both fail to score hits.
(b) Determine
i.
P(B/A)
ii.
P(A/ )
if A and B are events with P(A)=1/3, P(B)=1/4, P(AUB)=1/2.
9. A class has 10 boys and 5 girls. Three students are selected at random one after the other. Find
the probability that:
i.
First two are boys and third is girl
ii.
First and third of same sex and second is of opposite sex.
10. A box contains n tickets marked 1 through n. two tickets are drawn without replacements.
Determine the probability that the number on the tickets are consecutive integers.
11. If A and B are any two arbitrary events of the sample space the prove that P(AUB)= P(A)+P(B)P(A B).
OR
State and prove addition theorem of probability.
12. A ten digit number is formed using the digits 0-9, every digit being used only once. Find the
probability that the number is divisible by 4.
13. A and B throw alternately with a pair of dice. One who first throws a total of nine wins. What are
their respective chances of winning if A starts the game?
14. Three boxes, practically indistinguishable in appearance have two drawers each. Box 1 contains
a gold coin in one and silver coin in each drawer, box 2 contains a gold coin in each drawer and
box 3 contains a silver coin in each drawer. One box is chosen at random and one of its drawers
is opened at random and a gold coin is found. What is the probability that the other drawer
contains a coin of silver?
15. A and B throw alternatively with a pair of ordinary dice. A wins if he throws 6 before B throws 7
and B wins if he throws 7 before A throws 6. If A begins, show that his chance of winning is
30/61.
16. Suppose 5 men out of 100 and 25 women out of 10,000 are color blind. A color blind is chosen
at random. What is the probability of the person being male?
( Assume male and female to be in equal numbers)
17. Cards are dealt one by one from a well shuffled pack until an ace appears. Find the probability
that exactly n cards are dealt before the ace appears.
18. In a factory, machine A produces 40% of the output and machine B produces 60%. On the
average 9 items in 1000 produced by A are defective and 1 item in 250 produced by B is
defective. An item is drawn at random from a day’s output is defective. What is the probability
that it was produced by A or B?
19. Determine the probability for each of the following events:
i.
A non defective bolt will be found if out of 600 bolts already examined 12 were
defective.
20. Two digits are selected at random from the digits 1 through 9.
i. If the sum is odd, what is the probability that 2 is one of the numbers selected?
ii. If 2 is one of the digits selected, what is the probability that the sum is odd?
21. If
are n events then prove that
P(
)≥
( )-(n-1)
22. Companies
produces 30%, 45%,25% of the cars resp. it is know that 2%,3%,2% of these
cars produced from
i.
ii.
are defective.
What is the probability that a car purchased is defective?
If a car purchased is found to be defective, what is the probability that this car is
produced by the company ?
23. Two marbles are drawn in succession from a box containing 10 red, 30 white, 20 blue and 15
orange marbles, with replacement being made after each drawing. Find the probability that
i.
Both are white.
ii.
First is red and second is white.
24. A businessman goes to hotels X , Y, Z 20%, 50%, 30% of the time resp. It is known that 5%, 4%,
8% of the rooms in X,Y, Z hotels have faulty plumbing. What is the probability that
businessman’s room having faulty plumbing is assigned to hotel Z?
25. For any three arbitrary events A,B,C. prove that P(AUBUC)=P(A)+P(B)+P(C)-P(A B)-P(B C)P(C A)+P(A B C).
26. In a certain town 40% have brown hair, 25% have brown eyes and 15% have both brown hair
and brown eyes. A person is selected at random from the town,
i. If he has brown hair, what is the probability that he has brown eyes also?
ii. If he has brown eyes, determine the probability that he does not have brown hair.
27. Define random experiment, sample space, event and mutually exclusive events. Give examples
of each.
28. Box A contains 5 red and 3 white marbles and box B contains 2 red and 6 white marbles. If a
marble is drawn from each box, what is the probability that they are both of the same color?
29. Prove that
i. P( )=1-P(A)≤1.
ii. P(B)≤P(A) when B C A.
*********************UNIT 2 FAQ*********************
1. A continuous random variable X has a pdf given by,
F(x)=
; x≥0, λ>0
=0;
otherwise.
Determine the constant k, obtain the mean and variance of X.
2. For the continuous random variable X whose pdf is given by
F(X)= cx(2-x)
if 0≤ x <2
=0
otherwise.
Find c, mean and variance of X.
3. If X is a continuous random variable with pdf given by
F(X)= kx
if 0≤ x <2
=2k
if 2≤ x <4
= -kx+6k if 4≤ x <6
Find k and mean value of X.
4. If X and Y are discrete random variables and K is a constant then prove that
i.
E(X+K)=E(X)+K
ii.
E(X+Y)=E(X)+E(Y).
5. For the continuous probability function F(X)=
i.
ii.
iii.
k
Mean
Variance.
when X≥0,find
*********************UNIT 2*********************
1. If X is a continuous random variable and k is a constant then prove that
i.
Var(X+K)= Var(X)
ii. Var(KX)= Var(X)
2. Calculate expectation and variance of x, if the probability distribution of the random variable x is
given by
X
-1
0
1
2
3
f
0.3
0.1
0.1
0.3
0.2
3. Let X denote the minimum of thee two numbers that appear when a pair of dice is thrown once.
Determine the
i.
Discrete probability distribution
ii.
Expectation
iii.
Variance
4. If a random variable has the probability density f(x)= 2
for x>0= 0 for x £ 0. Find the
probabilities that it will take on a value
i.
Between 1 and 3
ii.
Greater than 5
5. Let X denote the number of heads in a single toss of 4 fair coins. Determine
i.
P(X<2)
ii.
P(1<X≤3)
6. Let F(x) be the distribution function of a random variable X given by,
F(x)= c
when 0≤x<3
=1
when x≥3
=0
when x<0.
If P(X=3)= 0,determine,
i.
c
ii.
The density function
iii.
Mean
iv.
P(X>1).
7. Let f(x)= 3 , when 0≤ x ≤1 be the probability density function of a continuous variable X.
determine ‘a’ and ‘b’ such that
i.
P(X≤a)= P(X>a)
ii.
P(X>b)= 0.05
OR
A continuous random variable X has a probability density function
F(x)= 3
=0
0≤ x <1
otherwise.
Find a and b such that,
i. P[X≤a]
ii. P[X>b]= 0.05.
8. A sample of 4 items is selected at random from a box containing 12 items of which 5 are
defective. Find the expected number of defective items.
9. If 3 cars are drawn from a lot 6 cars containing 2 defectives cars, find the probability distribution
of the number of defective cars.
10. For the discrete probability distribution.
X
0
1
2
F
0
k
2k
i. K
ii. Mean
iii. Variance
iv. Smallest value of x such that P(X≤x)>1/2.
3
2k
4
3k
11. A continuous random variable X has a pdf given by,
F(x)=
; x≥0, λ>0
=0;
otherwise.
Determine the constant k, obtain the mean and variance of X.
12. For the continuous random variable X whose pdf is given by
F(X)= cx(2-x)
if 0≤ x <2
=0
otherwise.
Find c, mean and variance of X.
13. Let X have pdf
F(x)= (x+1)/2
if -1< x <1,
=o
otherwise.
Find the mean and standard deviation.
14. If X is a continuous random variable with pdf given by
F(x)= kx
if 0≤ x < 2
= 2k
if 2≤ x < 4
5
6
7
7
+k
= -kx + 6k if 4≤ x <6
Find k and mean value of X.
15. For the discrete probability distribution.
x
0
f
0
Determine
i. K
ii. Mean
iii. Variance.
1
K
2
2k
3
2k
4
3k
5
6
7
+k
16. If X and Y are discrete random variables and k is a constant then prove that
i.
E(X+K)=E(X)+K
ii.
E(X+Y)=E(X)+E(Y)
17. For the continuous probability function F(X)=
iv.
v.
vi.
when x≥0,find
K
Mean
Variance.
18. If 3 cars drawn from a lot of 6 cars containing 2 defective cars, find the probability distribution of
the number of defective cars.
19. Define random variable, discrete probability distribution, continuous probability distribution and
cumulative distribution. Give an example of each.
20. A continuous variable X has the distribution function
F(x)= 0
if x≤ 1
=k
if 1< x ≤3
=1
if x>3.
Find
i.
K
ii.
The probability density function of x.
*********************UNIT 3 FAQ*********************
1. Show that if p is small and m is large, then the binomial distribution B(n,p) is approximated by
the poison distribution.
OR
Define poison distribution and find its variance and the mean.
2. Find the mean of the normal distribution.
OR
Find the arithmetic mean of the normal distribution.
3. In eight throws of a die 5 or 6 is considered a success. Find the mean number of the success and
the standard deviation.
4. In 256 sets of 12 tosses of a coin, in how many cases one can expect 8 heads and 4 tails?
5. Out of 800 families with 5 children each, how many would you expect to have
iii. 3 boys
iv. Either 2 or 3 boys.
6. 20% of items produced from a factory are defective. Find the probability that in a sample of 5
chosen at random.
i.
None is defective
ii. One is defective
iii. P(1<x<4).
7. If a poison distribution is such that P(x=1). = P(x=3). Find
i.
ii.
iii.
P(x≥1)
P(x≤3)
P(2≤ x ≤5)
8. A sales tax office has reported that the average sale of the 500 business that he has to deal with
during a year is Rs. 36,000 with a standard deviation of 10,000. Assuming that the sales in these
business are normally distributed. Find
i.
The number of business as the sales of while are Rs. 40,000/ii. The percentage of business the sales of while are likely to range between Rs. 30,000/and Rs. 40,000/9. Average number of accidents on any day on a national highway is 1.8. determine the probability
that the number of accidents are
i.
At least one
ii. At most one
10. If z is a normal variate, find,
i.
To the left of z= -1.78
ii. To the right of z= -1.45
iii. Corresponding to -0.80≤ z ≤1.53
iv. To the left of z= -2.52 and to the right of z= 1.83.
*********************UNIT 3*********************
1. The probability of a man hitting a target is 1/3.
i.
If he fires 5 times, what is the probability of his hitting the target at least twice?
ii.
How many times must he fire so that the probability of his hitting the target at least
once is more than 90%?
2. The average number of phone calls/minute coming into switch board between 2 p.m and 4 p.m
is 2.5. Determine the probability that during one particular minute there will be
i.
4 or fewer
ii.
More than 6 calls.
3. The mark obtained in mathematics by 1000 students is normally distributed with mean 78% and
standard deviation 11%. Determine
i.
How many students got marks above 90%
ii.
What was the highest mark obtained by the lowest 10% of the student
iii.
Within what limits did the middle of 90% of the students lie.
4. Determine the probability of getting 9 exactly twice in 3 throws with a pair of fair dice.
5. Suppose the weights of 800 male students are normally distributed with mean µ= 140 pounds
and standard deviation 10 pounds. Find the number of students whose weights are
i.
Between 138 and 148 pounds
ii.
More than 152 pounds.
6. 2% of the items of a factory are defective. The items are packed in boxes. What is the probability
that there will be
i.
2 defective items
ii.
At least three defective items?
7. The marks obtained in statistics in a certain examination found to be normally distributed. If
15% of the marks of the students ≥60, 40%< 30% marks, find the mean and standard deviation.
8. Show that if p is small and m is large, then the binomial distribution B(n, p) is approximated by
the poison distribution.
9. Find the mean and standard deviation of a normal distribution in which 7% of items are under
35 and 89% are under 63.
10. Fit the binomial distribution to the following data.
x
0
1
2
3
4
5
f
2
14
20
34
22
8
11. Average number of accidents on any day on a national highway is 1.8. determine the probability
that the number of accidents are
i.
At least one
ii.
At most one.
12. If z is a normal variate, find,
v. To the left of z= -1.78
vi. To the right of z= -1.45
vii. Corresponding to -0.80≤ z ≤1.53
viii. To the left of z= -2.52 and to the right of z= 1.83.
13. Fit the binomial distribution to the following data
x
0
1
2
3
4
5
f
38
144
342
287
164
25
14. Fit the poison distribution to calculate the theoretical frequencies for the following data.
x
0
1
2
3
4
f
109
65
22
3
1
15. Find the mean of the normal distribution.
OR
Find the arithmetic mean of the normal distribution.
16. Four coins are tossed 160times. The number of times x heads occur is given below.
x
0
1
2
3
4
No. of items
8
34
69
43
6
Fit a binomial distribution to this data on the hypothesis that coins are unbiased.
17. If x is a poison variate such that p(x=0)=p(x=2)+ 3p(x=4), find
i.
The mean of x
ii.
P(x≤2).
18. Prove that the mean= mode= median for a normal distribution.
19. If X is a poison variate such that p(x=0)= p(x=1), find p(x=0) and using recurrence formula find
the probabilities at x= 1,2,3,3 and 5.
20. In a sample of 1000 cases, the mean of a certain test is 14 and standard deviation is 2.5.
assuming the distribution to be normal, find
i.
How many students score between 12 and 15?
ii.
How many score above 18?
iii.
How many score below 8?
21. The mean and variance of a binomial distribution are 4 and 4/3 resp. find P(X≥1).
22. The marks of 1000 students in a university are found to be normally distributed with mean 70
and standard deviation 5. Estimate the number of students whose marks will be,
i.
Between 60 and 75
ii.
More than 75
iii.
Less than 68.
23. Let X be a random variable with E(X)= µ, standard deviation(X)= σ. If k>0 prove that
P{|X-µ|≥ kσ}≤ 1/ .
24. Show that the central moments of the binomial distribution satisfy the relation
= pq[nr
+ d / dp].
25. If X is normally distributed with mean 8 and standard deviation 4, find
i.
P(5≤ X ≤10)
ii.
P(10≤ X ≤15)
iii.
P(X≤15).
26. Ten coins are thrown simultaneously. Find the probability of getting at least seven heads.
27. The mean and variance of a binomial variable X with parameters n and p are 16 and 8. Find
P(X≥1) and P(X>2).
28. In eight throws of a die 5 or 6 is considered a success. Find the mean number of the success and
the standard deviation.
29. In a binomial distribution consisting of 5 independent trials, probabilities of 1 and 2 success are
0.4096 and 0.2048 resp. find the parameter p of the distribution.
30. In 256 sets of 12 tosses of a coin, in how many cases one can expect 8 heads and 4 tails?
31. Out of 800 families with 5 children each, how many would you expect to have
i.
3 boys
ii.
Either 2 or 3 boys.
32. 20% of items produced from a factory are defective. Find the probability that in a sample of 5
chosen at random.
iv. None is defective
v. One is defective
vi. P(1<x<4).
33. A distributer of bean seeds determines from extensive tests that 5% of large batch of seeds will
not germinate. He sells the seeds in packets of 200 and guarantees 90% germination. Determine
the probability that a particular packet will violate the guarantee.
34. Show that the mean deviation from the mean equals (approximately) to 4/5 of standard
deviation for normal distribution.
35. Find the maximum n such that the probability of getting no head in tossing a coin n times is
greater than 1.
36. Suppose 2% of the people on the average are left handed. Find
i.
The probability of finding 3 or more left handed
ii.
The probability of finding ≤1 left handed.
37. The mean and standard deviation of a normal variate are 8 and 4 resp. find
i.
P(5≤ x ≤10)
ii.
P(x≥5).
38. Find the probability that at most 5 defective components will be found in a lot of 200 it
experience. Shows that 2% of such components are defective. Also find the probability of more
than five defective components.
39. Write the importance of normal distribution.
40. If the mean and standard deviation of normal distribution are 70 and 16, find P(38)< x <46.
41. If a poison distributed is such that P(x=1). = P(x=3). Find
42.
43.
44.
45.
46.
i.
P(x≥1)
ii.
P(x≤3)
iii.
P(2≤ x ≤5).
A sales tax officer has reported that the average sale of the 500 business that he has to deal
with during a year is Rs. 36,000 with a standard deviation of 10,000. Assuming that the sales in
these business are normally distributed ,find
i.
The number of business as the sales of while are Rs. 40,000/ii.
The percentage of business the sales of while are likely to range between Rs. 30,000/and Rs. 40,000/-.
Assume that 50% of all engineering students are good in mathematics. Determine the
probabilities that among 18 engineering students
i.
Exactly 10
ii.
At least 10
iii.
At most 8
iv.
At least 2 and at most 9, are good in mathematics.
A student takes a true false examination consisting of 8 questions. He guesses each answer. The
guesses are made at random. Find the smallest value of n that the probability of guessing at
least n correct answers is less than (1/2).
Using the recurrence formula find the probabilities when x=0, 1,2,3,4 and 5. If the mean of
poison distribution is 3.
If the masses of 300 students are normally distributed with mean 68 kgs and standard deviation
3 kgs. How many students have masses.
i.
Greater than 72 kg
ii.
Less than or equal to 64 kg
iii.
Between 65 and 7 kg inclusive.
*********************UNIT 4 FAQ*********************
1. A population consists of five numbers 2,3,6,8,11. Consider all possible samples of size two which
can be drawn without replacement from the population. Find,
i.
The mean of the population
ii. Standard deviation of the population.
iii. The mean of the sampling distribution of means
iv. The standard deviation of the sampling distribution of means.
2. A random sample of size 100 is taken from an infinite population with mean 76 and variance
256. Find the probability that the mean of the sample is in the interval (75,78).
3. The diameter of rotor shafts in a lot has a mean of 0.249 inch and a standard deviation of 0.003
inch. The inner diameter of bearings in another lot has a mean of 0.255 inch and a standard
deviation of 0.002 inch.
i.
What are the mean and the standard deviation of the clearances between shafts and
the bearings selected from these lots?
ii. If a shaft and a bearing are selected at random, what is the probability that the shaft will
not fit inside the bearing? (Assume that both dimensions are normally distributed).
4. A random sample of size 144 is taken from an infinite population having the mean 75 and the
variance 225. What is the probability that will lie between 72 and 77?
*********************UNIT 4*********************
1. The mean of certain normal population is equal to the standard error of the mean of the
samples of 64 from that distribution. Find the probability that the mean of the sample size 36
will be negative.
2. A normal population has a mean 0.1 and standard deviation of 2.1. Find the probability that the
mean of simple sample of 900 members will be negative.
3. What is the probability that will be between 75 and 78 if a random sample of size 100 is taken
from an infinite population has mean 76 and variance 256?
4. If the population is 3,6,9,15,27
i. List all possible samples of size 3 that can be taken without replacement from the finite
population.
ii. Calculate the mean of each of the sampling distribution of means.
iii. Find the standard deviation of sampling distribution of means.
5. A population consists of five numbers 2, 3, 6,8,11. Consider all possible samples of size two
which can be drawn without replacement from the population. Find,
6.
7.
8.
9.
10.
11.
12.
v. The mean of the population
vi. Standard deviation of the population.
vii. The mean of the sampling distribution of means
viii. The standard deviation of the sampling distribution of means.
A population random variable x has mean 100 and standard deviation 16. What are the mean
and standard deviation of the sample mean for random samples of size 4 drawn with
replacement.
Let S={1,5,6,8}. Find the probability distribution of the sample mean for random sample of size 2
drawn without replacement.
Determine the probability that the sample mean area covered by a sample of 40 of 1 litre paint
boxes will be between 510 to 520 square feet given that 1 litre of such paint box covers on the
average 513.3 square feet with standard of 31.5 square feet.
Define the statistics t and F and write down their sampling distributions. State the important
assumptions in respect of them.
Two independent random samples of size 8 and 7 gave variances 4.2 and 3.9 resp. do you think
that such a difference has probability less than 0.05. Justify your answer.
A random sample of size 100 is taken from an infinite population with mean 76 and variance
256. Find the probability that the mean of the sample is in the interval (75, 78).
If two independent random samples of sizes
are taken from a normal
population, what is the probability that the variance of the first sample will be at least 4 times as
large as the variance of the second sample.
13. The diameter of rotor shafts in a lot has a mean of 0.249 inch and a standard deviation of 0.003
inch. The inner diameter of bearings in another lot has a mean of 0.255 inch and a standard
deviation of 0.002 inch.
iii. What are the mean and the standard deviation of the clearances between shafts and
the bearings selected from these lots?
iv. If a shaft and a bearing are selected at random, what is there probability that the shaft
will not fit inside the bearing? (Assume that both dimensions are normally
distributed).
14. A random sample of size 144 is taken from an infinite population having the mean 75 and the
variance 225. What is the probability that will lie between 72 and 77?
15. Two random sample of sizes 15 and 25 are taken from a N(µ,
ratio of the sample variances does not exceed 2.28.
). Find the probability that the