Stats Practice material

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PROBABILITY DISTRIBUTION
1. The probability distribution of a random variable x is defined as follows:
x 1 2 3
P(x) K 4K 9K
Find P(x ≥ 2)
a.
1
14
b.
13
14
c.
4
10
d.
9
14
2. If x is a discrete random variable which takes the value 0, 1, 2 & P(x=0) =
144
169
,
P(x=1) =
1
169
. Then the value of P(x=2) = ………
a.
24
169
b.
25
169
c. 0 d. 1
3. A random variable x has the following probability distribution:
X 0 1 2 3 4 5
P(x) 0 3K 2K 3K2
7 K2
7 K2
+ K
then the value of K:
a.
1
10
b. -1 c. 0 d. 1
4. A discrete random variable x follow uniform distribution & takes the values 10, 11, 12,
13. Find the P(x > 11)
a.
1
11
b.
4
11
c.
1
2
d.
3
4
5. The binomial distribution is a
a. Continuous distribution b. Discrete distribution
c. Both (a) & (b) d. None of the above
6. When a coin is tossed for 10 times then ……….. distribution can be used.
a. poisson b. binomial c. normal d. none of
these
7. The expected value of the number of successes of a binomial distribution is equal to:
a. Zero
b. the probability of success
c. the number or trials (n)
d. the product of the number or trials (n) & the probability of success
8. The parameters of the binomial distribution are
a. µ, 𝜎2 b. p, q c. 0, 1 d. n, p
9. In a binomial distribution
a. mean is less than variance b. mean is equal to variance
c. mean is greater than variance d. none of these
10. In a binomial distribution with the probability of success equal to the probability of failure.

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a. mean is less than variance b. mean is equal to variance
c. mean is greater than variance d. none of these
11. For a binomial distribution which of the following is correct?
a. mean = 7, variance = 16 b. mean = 5, variance = 9
c. mean = 4, variance = 12 d. mean = 6, variance = 2
12. The S.D of the binomial distribution is:
a. Square of np b. Square root of np
c. Square of npq d. Square root of np(1 – p)
13. In a binomial distribution if n = 4, p = 2/3 then variance is:
a. 1/9 b. 8/9 c. 8/3 d. 3/8
14. In a binomial distribution if np = 5 & npq = 2.25 then p is equal to
a. 0.45 b. 0.55 c. 1 d. 0
15. A shop has 5 employees. The owner has observed that there is a 0.4 chance of any one
employee being late & that they arrive independently of one another. Find the probability
that 3 employees will be on time on a particular day.
a. 0.3456 b. 0.2304 c. 0.216 d. 0.064
16. 90% of ships safely reach to the destination. If 400 ships start the journet. Find the mean
& variance of numbers of ships safely reaching the destination.
a. 40, 6 b. 360, 6 c. 40, 36 d. 360, 36
17. It is known that a patient if treated by Dr. A, the probability of no recovery is 0.2. Find
S.D of no. of patients who have recovered if 100 patients are treated by Dr. A.
a. 16 b. 20 c. 80 d. 4
18. A balanced dice is tossed 5 times. If the event to get an odd number is called a success.
Find the probabilityt of getting exactly 3 successes.
a. 5/32 b. 250/7776 c. 5/16 d. None of
these
19. Team A & B play a game. Probabilityt that team A wins the game is 0.7, what is the
probabilityt that team B wins 3 out of 4 games?
a. 0.0756 b. 0.756 c. 0.4116 d. 0.1764
20. There are 40 employees in a company, the probability that the height of an employee is
less than 180 cm is ¼. A random sample of 5 employees is taken one by one with
replacement from the company. What is the probability that 4 employees have a height
that is more than 180 cms?
a. 15/1024 b. 135/341 c. 405/1024 d. None of
these

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21. In past two playeers A & B have played 20games of which 10 are won by A, 6 are won
by B & 4 ended in a tie. Now in a tournament they are to play 4 games. What is the
probability that 2 games will end in a tie?
a. 3/8 b. 0.2646 c. 96/625 d. 5/8
22. Out of 320 families with 5 childern each, how many families would be expected to have 2
boys? (Assume boys & girls are equiprobable)
a. 160 b. 10/32 c. 2.5 d. 100
23. If 20% of the bulbs manufactured by a company are defective. What is the average
number of defectrive bulbs in a batch of 400 bulbs?
a. 320 b. 80 c. 64 d. 8
24. If 20% of the bulbs manufactured by a company are defective. What is the variance of
number of non-defectrive bulbs in a batch of 500 bulbs?
a. 80 b. 8.94 c. 100 d. 400
25. The mean & variance of a binomial distribution are 10 & 5 respectively. What is the
probability of failure in the distribution?
a. 2 b. ½ c. 20 d. none of
these
26. If x is a binomial variable with total no. of trails (n) is 16. If its is symmetric distribution.
Then mean of this distribution.
a. 16 b. 8 c. 0 d. not exist
27. For a binomial distribution, total no. of trails (n) are 9 & P(x=4) = 2P(x=5). Find
probability of success is:
a.
1
3
b.
1
2
c. 1 d.
4
5
28. x is a binomial random & if n = 4, P(x=2) =8/27, then variance is
a. 8/9 b. 9/8 c. 3/8 d. 8/3
29. For a binomial distribution P(x) = 10Cx(0.5)x
(0.5)10-x
the mean value is
a. 5 b. 4 c. 15 d. 10
30. X is a binomial variate with n = 200. What is the mean of x if the distribution is
symmetric?
a. 0 b. 50 c. 100 d. 1
31. If P(x) = px
q1-x
, x = 0.1 & q = ¼ then P(x=0) is
a. ¾ b. ¼ c. 1 d. None of
these

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32. For a binomial distribution coefficient of variation is
100
3
& probability of success is
1
3
. Find
total number of trails (n).
a. 8 b. 33 c. 100 d. 18
33. For a binomial variate n = 6 & P(x=4) : P(x=3) = 3 : 8. Find P(x > 0).
a. 0.375 b. 0.9973 c. 0.002743 d. 0.625
34. If 2P(x=3) = 3P(x=2). What is the value of p?
a. 8/17 b. 9/17 c. 8/9 d. 1
35. How many tosses of a coin are needed, so that the probability of getting atleast one
head is 0.9375.
a. 2 b. 10 c. 8 d. 4
36. For a binomial distribution, total number of trials are 10 & probability of getting 5 success
is double the probability of getting 4 success. Find p.
a. 5/8 b. 3/8 c. 3/5 d. 2/5
37. Pakistan cricket team losses twice as often as it wins. What is the probability that in next
5 matches, Pakistan wins 3 matches?
a. 80/243 b. 10/32 c. 11/16 d. 40/243
38. The sum & the product of the mean & variance of a binomial distribution are 16 &
48respectively. What is the probability of success?
a. 0 b. 2/3 c. 1/3 d. none of
these
39. The sum & differnce of the mean & variance of a binomial distribution are 95 & 5
respectively. Find total number of trials (n).
a. 50 b. 45 c. 450 d. 500
40. The probability of reaction of grug is p. The drug is given to 4 patients. If the probabilty
that 2 patients develop reaction is 8/27. Find the value of p.
a. Only 1/3 b. Only 2/3 c. Only 8/27 d. Both ‘a’ & ‘b’
41. For a binomial probability distributionif n = 12 &
𝑃(𝑥=2)
𝑃(𝑥=3)
=
3
4
, what is the probability of
failure?
a. 2/7 b. 5/7 c. 2/3 d. 3/4
42. How many tosses f a coin are needed, so that the probability of getting atleast one head
is 0.96875?
a. 3 b. 4 c. 2 d. 5
43. A random variable Y has the following probablity distribution:
Y 20 40 60 80 100

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Probability 0.15 0.20 0.30 0.25 0.10
If 15 values of Y have been observed. Find average & variance number of observations
Y such that 40 ≤ Y ≤ 80.
a. 11.25, 2.81 b. 15, 3.87 c. 13.5, 1.35 d. 14.25, 0.71
44. The binomial distribution be approximated to Poisson distribution, if the probability of
success (p) is ……. & the number of trials (n) is ………
a. large, small b. small, large c. small, small d. large, large
45. In Poisson distribution
a. mean = variance b. mean > variance
c. mean < variance d. mean = standard deviation
46. Poisson distribution is known as ………. Distribution.
a. uni-parameter b. bi-parameter
c. tri-parameter d. none
47. If the mean of a poisson distribution is 1, what is the P(atleast one)?
a. 1 – e-1
b. e-1
c. e-2
- 1 d. e
48. A company provides ‘Dial a Car’ services for which they have certain number of cars.
The owner by his past experience knows that the average demand of cars per day is 2.
What is the probability that on a particular day not more than 3 cars are in use?
a. 0.1436 b. 0.1804 c. 0.2707 d. 0.8564
49. At share broker’s office number of phone calls between 3 p.m & 5 p.m is 30. Find the
probability that during a particular time of 20 minutes there are exactly 3 phone calls.
a. 0.2240 b. 0.8596 c. 0.1404 d. 0.03889
50. If X is a Poisson variate such that P(x=1) = P(x=2). Find mean.
a. 0 b. 1 c. 2 d. 3
51. If X is a Poisson variate such that P(x=1) = P(x=2). Find P(x=0)
a. 0.1353 b. 0.2707 c. 0.3679 d. 0.0498
52. If x is a Poisson variate such that P(x=0) = P(x=1) = K. find the value of K.
a. 0.1353 b. 0.2707 c. 0.3679 d. 0.0498
53. The manufacturer of pencil cells knows that 0.5% of cells are defective. He claims that in
a box not more than 2 of cell are defective. Pencil cells are packed in boxes of 200. Find
the probability that a box selected at random will not satisfy the manufacturer’s claim.
a. 0.9198 b. 0.998 c. 0.0802 d. 0.002
54. In a Poisson distribution, coefficient of variation is 57.67%. find the P (x=0)
a. 0.3679 b. 0.9502 c. 0.6321 d. 0.0498

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55. A random variable x follows Poisson distribution, such that P(x = a) = P(x = a – 1). What
is the standard deviation of the distribution?
a. a – 1 b. a c. a + 1 d. √ 𝑎
56. In a Poisson distribution if P(x=0) = P(x=1) = a, the value of a is
a. 1 b. 0 c. e d. 1/e
57. If standard deviation of a Poisson distribution is m, then its mean is
a. √ 𝑚 b. m c. m2
d. none of
these
58. x is a Poisson vdistribution with parameter 0.5, then P(x >1) =
a. 1 – 15e0.5
b. (2 – 3e-0.5
)/2 c. 1 – e-0.5
d. 1.5e-0.5
59. If for a Poisson distribution of x, P(4) =
1
3
P(2), what is the standard deviation of x?
a. 2 b. 4 c. 2/3 d. None of
these
60. In a Poisson distribution, P(x = 0) is 10%. What is the mean of the distribution?
a. 2.3026 b. 10 c. e-10
d. 0.4343
61. If x is Poisson variable such that P(x=2) = 9P(x=4) + 90P(x=6). Find the mean &
varaince of x.
a. 4 b. -1 c. 2 d. 1
62. The number of accidents in a year attributed to the state transport bus driver follows
Poisson distribution wiyh mean 3. Out of 1000 bus drivers, how many drivers had more
than 1 accident in a year.
a. 199 b. 801 c. 736 d. None of
these
63. If 5% of the electric bulbs manufactured by a company are defective. Find the probability
that in a sample of 100 bulbs, 5 are bulbs are defective.
a. 0.1723 b. 0.1623 c. 0.1823 d. 0.1923
64. In the manufacturing of electric fuses, it is known that 0.5% of the fuses are defective.
The fuses are sold in boxes of 100 & it is guaranteed that not more than 2 will be
defective in a box. What is the probability that a box will meet this guarantee?
a. 0.9098 b. 0.07581 c. 0.0144 d. 0.9856
65. In office switch board receives 40 phone calls between 1 p.m & 4 p.m. what is the
probability that during a particular 5 minutes interval one phone call is recevied?
a. 0.1766 b. 0.3296 c. 0.8025 d. 0.3659

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66. If the mean & S.D of Poisson distribution is m & √ 𝑛 respectively, then
a. m = √ 𝑛 b. m = n c. √ 𝑚 = n d. m = n2
67. Normal distribution is a limiting case of binomial distribution if
a. n → ∞, p or q are not very small b. n → ∞
c. p & q are small d. none of these
68. The toral area under the normal curve is
a. 0.5 b. 0 c. 0 to 1 d. 1
69. Normal distribution is
a. Positively skewed b. Symmetric
c. Negatively skewed d. None of these
70. In normal distribution, relation between mean, median & mode is
a. mean > median > mode b. mean < median < mode
c. 2mean = median = mode d. mean = median = mode
71. The daily wages of a workers selected from normal distribution with mean Rs.150 & a
standard deviation Rs.20. what is the proportion of workers having wages less than
Rs.160?
a. 0..6715 b. 0.3285 c. 0.1915 d. 0.8085
72. A sales tax officer has noted that the average sales during a year of 5000 firms is
Rs.100,000 with a standard deviation of Rs. 20,000. Assuming that the sales in these
firms are normally distributed, find the probability of firms having sales between
Rs.80,000 & Rs.110,000.
a. 0.4672 b. 0.0328 c. 05328 d. 0.6915
73. The mean & standard deviation of the height of a group of 500 persons are 105 cm & 10
cm respectively. The distribution of the height of the persons is assumed to be normally
distributed. Whati is the maximum height of tallest 20% persons of the group?
a. 100 b. 111.2 c. 92.4 d. 113.4
74. The mean & standard deviation of the height of a group of 500 persons are 105 cm & 10
cm respectively. The distribution of the height of the persons is assumed to be normally
distributed. Whati is the maximum height of shortest 10% persons of the group?
a. 100 b. 111.2 c. 92.4 d. 113.4
75. The mean & standard deviation of a normal distribution are 20 & 2 respectively. Find the
range of 62% of the observations located in the middle of the distribution.
a. 18.24 to 21.76 b. 15.73 to 21.76
c. 15.73 to 20.11 d. 18.24 to 20.11

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76. The income distribution of 10,000 persons was found to be normal with standard
deviation Rs.50. If 9987 persons have their income exceeding Rs.600. Find mean
income of the distribution.
a. 550 b. 600 c. 650 d. 700
77. The marks obtained by students in examination of 100 marks is normally distributed. If
15.87% of students score marks more than 60 & 2.28% of students score marks less
than 36. What is the mean & variance of the distribution?
a. 52, 8 b. 52, 64 c. 60, 36 d. 60, 6
78. If 16% of the values in normal distribution are greater than 600 & coefficient of variation
is 20%. Find the mean & S.D of the distribution.
a. 400, 200 b. 450, 150 c. 500, 100 d. 650, 50
79. The limits for central 50% of the observations are respectively 18.64 & 31.32. Find
coefficient of variation of vthe distribution.
a. 59.51% b. 37.87% c. 37.28% d. 62.64%
80. The distribution of the weights of students of a college is normally distributed with mean
55 kg & S.D 5 kg. if 80 students have their weights more than 65 kg. Find the total
number of students in the college.
a. 3509 b. 3415 c. 3537 d. 3491
81. The mean & S.D of the normal distribution are 50 & 5 respectively. Find fourth decile.
a. 45.14 b. 46.63 c. 48.75 d. 52.63
82. The marks of the students in a certain examination are normally distributed with mean
marks as 70 & S.D marks as 20. If the authority wants only 40% of students to fail, than
what should be the passing marks?
a. 60 b. 65 c. 70 d. 75
83. A coin is tossed 500 times. Find the probability that the number of tails is between 230 &
245.
a. 0.2476 b. 0.25 c. 0.2897 d. 0.3146
84. If 25% of observations are less than 24.67 & 255 of observations are more than 56.33 in
normal distribution. What is the value of coefficeint of variation?
a. 172.56 b. 40.5 c. 550.84 d. 57.95
85. If 68% of observations in normal distribution are greater than 200 & coefficient of
variation is 30. Find the variance of the distribution.
a. 69.85 b. 141.84 c. 4879.023 d. 20119.71
86. A random variable x is normally distributed with mean is 100 & S.D is 4. If P(90 < x < K)
= 0.3. find K

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a. 100.89 b. 102.3 c. 99.11 d. 97.7
87. The marks obtained by students in an exam follow the normal distribution with mean 30
& S.D is 5. What are the percentage of students failing if passing marks is 17?
a. 47% b. 49.53% c. 30.15% d. 0.47%
88. The life of electric tubelights follow normal distribution with mean life of 3000 hours &
S.D of 200 hours. After how many hours 10% of tube lights will still be burning?
a. 2744 b. 3256 c. 3040 d. None of
these
89. The distribution of the weights of 700 students is assumed to be normal with mean 55 kg
& varince 49kg. what is ther range of weghts for middle 30% of students?
a. 36.13, 73.87 b. 49.12, 60.88 c. 52.31, 57.7 d. 13.84, 96.16
90. The distribution of the heights of students is assumed to be normal with mean 105 cm &
varince 100 cm2
. If 22 students have heights more than 122 cm. Find the total number of
students.
a. 51 b. 98 c. 493 d. None of
these
91. If in a normal distribution 4th
decile is 48.75 7 70th
percentile is 76.25. what are mean &
S.D of the distribution?
a. 50, 25 b. 50, 5 c. 100, 25 d. 100, 5
92. A random variable x is normally distrinbuted with mean 100 & S.D 10. For what value of
K, P(x > K) = 0.25?
a. 93.25 b. 102.5 c. 106.75 d. 97.5