1) A random variable X.find c
X : 1 2 3 4
P(x): c 2c 3c 4c
A) 0.2 B) 0.3 C) 0.1 D) 0.4
2) A random variable X has:
X: 0 1 2 3
P(x): 2/6 3/6 0/6 1/6 find mean and variance of X are:
A) 1,2 B) 1,3 C) 1,1 D) 3,1
3) Find k from the following:
X: 1 2 3 4
P(x): 2k 4k 3k k
A) 0.2 B) 0.3 C) 0.1 D) 0.5
4) If the range of a random variable X is (0, 1,3,.....) With P(X= k)= {(k+1)a/3ᵏ for k≥ 0, then a= ?
A) 4/9 B) 16/81 C) 5/9 D) 25/81
5) A random variable X has its range {0,1,2,3} and the Probabilities are given by: P(X=0)= 2k⁴, P(X=1)= 3k² - 5k³, P(X=2)= 2k - 3k², P(X=3)= 3k - 1. Find k.
A) 1/2 B) 1/3 C) 1/4 D) n
6) X: -3 -2 -1 0 1 2
P(X): 0.1 2k 3k 7k 0.2 0.1 Find mean
A) 0.25 B) 0.35 C) -0.25 D) -0.35
7) X: 2 4 6 8 10 12
P(X): 0.1 0.2 0.3 0.1 0.1 0.2 Find mean
A) 3 B) 5 C) 7 D) 9
8) A random variable X takes the values 0,1 and 2. if P(X=1)= P(X=2) and P(X=0)=0.4, then the mean of the random variable X is:
A) 1.1 B) 0.9 C) 0.8 D) 0.4
9) A random variable X takes the values 0,1, 2,3 and its mean is 1.3. If P(X=3)= 2P(X=1) and P(X=2)=0.3, then P(X=0)=
A) 0.40 B) 0.30 C) 0.25 D) 0.15
10) Theoretical distribution is a:
A) Probability distribution
B) Standard distribution
C) Random distribution. D) none
11) In discrete case the Probability of the entire space is:
A) 0.5 B) 1 C) 0 D) none
12) Binomial distribution is a:
A) Discrete Probability distribution
B) continuous Probability distribution
C) Both A and B D) Neither A nor B
13) An important discrete Probability distribution is:
A) Normal Distribution
B) Binomial distribution
C) Geometric Distribution D) none
14) Binomial distribution is symmetrical if:
A) p> q B) p=q C) p <q D) Binomial distribution is never symmetrical.
15) The Probability mass function of binomial distribution is given by:
A) f(x)= pˣ qⁿ⁺ˣ B) f(x)= ⁿCₓ Pˣ qⁿ⁻ˣ
C) f(x)= ⁿCₓ qˣ pⁿ⁻ˣ D) ⁿCₓ pⁿ⁺ˣ qˣ
16) In Binomial Distribution 'p' denotes probability of:
A) Success B) Failure
C) both A and B above. D) none
17) The important characteristic(s) of Binomial trials is
A) Trials are independent.
B) each trial is associated with just two possible outcomes.
C) trials are infinite
D) both A and B.
18) When there are fixed number of repeated trials of any experiment under identical conditions for which only one of the two mutually exclusive outcomes, success or failure can result in each trial then it involves:
A) normal distribution
B) Binomial distribution
C) poisson distribution D) N
19) if X is a variable with parameters n and p, then X can assume:
A) Any value between 0 and n, both inclusive.
B) any value between 0 and n.
C) any whole number between 0 and n, both inclusive.
D) any number between 0 and infinity.
20) The mean of Binomial Distribution is :
A) always more than its variance
B) always equals to its standard deviation.
C) always less than its variance
D) always equal to its veriance.
21) In Binomial distribution 'n' means.
A) number of trials of the experiment.
B) number of success.
C) the probability of getting success. D) none
22) The mean of a Binomial Distribution with parameter n and p is:
A) n(1-p) B) np(1-p) C) np D) none
23) When p= 0.5, the Binomial Distribution is:
A) asymmetrical B) symmetrical
C) Both A and B
D) Neither A nor B
24) when p= 0.1, the Binomial Distribution is skewed to the
A) right B) left
C) neither A nor be B
D) both A and B
25) A Binomial Distribution is:
A) never negatively skewed
B) never positively skewed
C) never symmetrical
D) symmetrical when p= 0.5
26) when p is larger than 0.5, the Binomial distribution is:
A) asymmetrical B) Symmetrical
C) both A and B D) Noneof A or B
27) For Binomial distribution, mean and mode
A) are always equal
B) are never equal
C) are equal when q= 0.50
D) do not always exist.
28) mean of Binomial Distribution is
A) no. B) npq C) neither A nor B
D) cannot be determined
29) Variance of Binomial distribution is
A) np B) npq C) nq D) PQ
30) when a coin is tossed 10 times, then it is a case of
A) Normal Distribution
B) poisson distribution
C) Binomial Distribution D) none
31) An example of a bi-parametric discrete probability distribution is:
A) binomial distribution
B) normal distribution
C) Poisson distribution
D) both A and B above
32) In Binomial Distribution if n is infinitely large, the probability p of occurrence of event is close to__ and q is close to___
A) 0,1 B) 1,1. C) 1, p D) none
33) The mean of a Binomial Distribution is 5 and standard deviation is 3
A) True B) False C) can't say D) n
34) The maximum value of the variance of a Binomial distribution with parameters n and p is:
A) n/p B) n/3 C) n/4 D) n/2
35) A random variable follows Binomial distribution with mean 2 and variance 1.2. then the value of n is:
A) 3 B) 5 C) 7 D) none
36) If in a B. D, np=9 and npq =2.25, then find q
A) 0.75 B) 0.25 C) 0.10 D) 0.45
37) The method usually applied for hitting a Binomial distribution is known as:
A) method of probability distribution
B) method of moments
C) method of least square
D) method of deviation
38) For n independent trials in Binomial Distribution, the sum of the powers of p and q is always n, whatever be the number of success.
A) True B)False C)both A and B d) n
39) An unbiased dice is tossed 500 times. The standard deviation of the number of 'sixes' in these 500 tosses is:
A) 15/6 B) 51/6 C) 50/6 D) 4/7
40) If in B. D mean= 20, S. D= 4, then p is:
A) 1/5 B) 2/5 C) 3/5 D) 4/5
41) If in B. D mean= 20, S. D= 4, then n is:
A) 119 B) 95 C) 100 D) 105
42) If in B.D , n= 4, p= 1/3, then variance is:
A) 3/8 B) 9/8 C) 8/9 D) 8/3
43) if neither P not Q is very small n sufficiently large, the Binomial Distribution is very closely approximated by ____ distribution
A) Poisson distribution
B) Geometric distribution
C) Normal distribution
D) all of the above.
44) The result of ODI matches between India and Pakistan follows
A) Binomial Distribution
B) normal distribution
C) Poisson distribution
D) geometric distribution.
45) If in B. D mean= 20, S. D= 4, then q is:
A) 5/4 B) 1/5 C) 4/5 D) 2/5
** 6 coins are tossed. Find the probability of getting
46) all heads.
A) 1/64 B) 15/64 C) 22/64 D) 5/16
47) no heads.
A) 15/64 B) 22/64 C) 5/64 D)1/64
48) Two heads.
A)15/64 B)22/64 C) 5/16 D)31/64
49) Three heads.
A)15/64 B)22/64 C)5/16 D)1/64
50) two or more heads.
A)15/64 B)22/64 C)5/16 D)57/64
51) At most two heads.
A)15/64 B)22/64 C)5/16 D) none
52) A student update the following results: For the binomial distribution mean=4, variance =3. comment on the accuracy of his results.
A) correct results, with n= 12
B) correct results, with n=16
C) wrong calculations, as q> 1
D) none of these
53) A student obtained the following results: for the Binomial Distribution mean =7, variance= 11. comment on the accuracy of his results.
A) correct results, with n=16
B) correct result as variance is greater than mean.
C) wrong calculations as variance can't be greater than mean.
D) none
54) X obtain the following results: the mean of a Binomial Distribution is 3 standard deviation is 2. comment on the accuracy of his calculations.
A) wrong calculations as variance cannot be more than mean.
B) correctly done
C) correct calculation with n= 100
D) nothing can be said
** Five coin are tossed simultaneously. What is the probability of
55) three heads
A)10/16 B)10/32 C)1/2 D) None
56) at least three heads
A) 1/3 B) 1/32 C) 1/16 D) 1/2
57) more than three heads
A) 5/16 B) 3/16 C) 7/16 D) n
58) A machine produces 2% defectives on an average. If 4 articles are chosen randomly. What is the probability that there will be exactly 2 defective articles?
A)0.229 B)0.115 C)0.235 D)0.452
59) The mean and standard deviation of a binomial distributions are 4 and √(8/3). The values of n and p are:
A) 12, 0.25 B) 12, 0.50
C) 12, 0.75. D) 16, 0.50
** For a Binomial Distribution the mean is 3 and the variance is 2 .
80) what is the probability of success of the said Binomial Distribution
A)0.50 B) 0.25 C) 0.33 D) 0.65
61) How many times the experiment was repeated?
A) 8 B) 9 C) 10 D) 25
62) what is the probability of failure of the experiment
A)0.50 B) 0.75 C) 0.67 D) 0.5
63) what is the probability that the variable assume the value 5
A)0.2048 B)0.3524 C)0.2456 D) 0.1024
64) For a Binomial distribution, mean is 20 and S. D is 4 .
A) what is the probability of success
A) 0.20 B) 0.30 C) 0.25 D) 0.35
65) what is the value of n
A) 110 B) 90 C) 85 D) 100
66) which of the following gives you the value of q
A) 0.65 B) 0.75 C) 0.80 D) 0.70
67) In a shooting competition, the probability of man hitting a target is 1/5. If he fires 5 times, what is the probability of hitting the target atleast twice?
A) 821/3125 B) 1024/3125
C) 2304/3125 D) none
*** Assume the probability that bomb dropped from an aeroplane will strike a target is 1/5. If 6 bombs are dropped. Find the probability that.
68) Exactly 2 will strike the target
A) 698/3125 B) 768/3125
C) 2304/3125 D) none
69) atleast 2 will strike the target
A) 4536/15625 B) 2596/3125
C) 6533/3125 D) none
70) For a Binomial Distribution, the mean and standard deviation are respectively 5 and 2. Find the probability of getting a non-zero value from the distribution.
A) (0.8)²⁵. B) 1 - (0.8)²⁰
C) 1 - (0.8)²⁵. D) none
** If the probability of a defective bulb be 1/5. Find the following of a Binomial distribution of defective bulb in a total of 400 bulbs:
71) The mean
A) 8 B) 80 C)10 D) 100
72) the standard deviation:
A) 81 B) 10 C) 16 D) 8
71) The overall percentage of failures in a certain examination is 60. What is the probability that out of a group of 6 candidates atleast 5 passed the examination?
A) (0.4)⁶. B) (0.4)³(0.6)³
C) (0.6)⁵ + (0.6)⁶. D) 4(0.4)⁵
74) 8 coins are tossed 250 times. What is the mean and standard deviation of the distribution thus formed?
A) 8, 2 B) 4, 2 C) 4, 1.41 D) none
75) In 10 independent throws of a defective die, the probability that even number will appear 5 times is twice the probability that the even number will appear 4 times. Find the probability that even number will not appear at all in 10 independent throws of the die.
A) (5/8)¹⁰ B) (3/8)¹⁰
C) (5/8)⁵(3/8)⁵ D) none
76) Suppose that the half the population of a town is a consumers of rice. 100 investigators are appointed to find out its truth. Each investigator interviews 10 individuals. How many investigator do you expect to report that three or less of the people interviewed are consumers of rice.
A) 19 B) 17 C) 21 D) 23
77) The incidence of occupational disease in an individual in such that the workmen have a 20% chance of suffering from it. What is the probability that out of 6 workmen, 4 or more will contact the disease?
A) 353/3125 B) 35/3125
C) 53/3125 D) none
*** Find the probability that in a family of 4 children there will be: (Assuming that the probability of a male birth is 1/2)
78) Atleast 1 boy.
A) 1/16 B) 1- 1/16 C) 1- 1/8 D) 1/8
79) Atleast 1 boy and 1 girl.
A) 1- 1/8 B) 1/8 C) 1/16 B) 1- 1/16
*** Out of 2000 families with 4 children each, how many would you expect to have:
80) Atleast 1 boy
A) 1750 B)1875 C) 750 D) 1000
81) Atleast 1 boy and 1 girl
A) 750 B)1000 C) 1875 D) 1750
82) Exactly 2 girls.
A) 750 B)850 C) 1050 D) 950
83) Four coins are tossed simultaneously, what is the probability of getting two heads and two tails.
A) 2/8 B) 5/8 C) 1- 5/8 D)1- 3/8
*** 6 fair coins are tossed simultaneously. Find the probability of getting:
84) exactly four heads.
A) 13/64 B) 15/64 C) 11/32 D) n
85) at least three heads.
A) 11/32 B) 15/64 C) 1/32 D) n
86) 25% of the inhabitants in a large town are pay bespectacled. what is the probability that a randomly selected group of 6 inhabitants will include at the most 2 bespectacled person.
A) 1701/2408 B) 1809/2048
C) 1701/2048. D) 2048/2480
87) A random variable follows Binomial Distribution with mean 4 and standard deviation √2. find the probability of assuming non zero value of the variant.
A) (0.5) B) (0.5)⁸. C) 1- (0.5)⁸ D) n
88) Assuming the half the population is vegetarian and each of 128 investigators take a sample of 10 individuals to see whether they are vegetarian. How many investigators would you expect to report that two people or less vegetarians.
A) 5 B) 7 C) 9 D) 11
** The rainfalls on an average 12 days in a month after 30 days. find the probability that a given week:
89) the first four days are fine, remaining wet
A) (0.6)⁴ (0.4)³. B) 35.(0.6)⁴ (0.4)³
C) 35(0.6)³ (0.4)⁴. D) (0.6)³ (0.4)⁴
90) three days are raining.
A) (0.6)⁴ (0.4)³. B) 35.(0.6)⁴ (0.4)³
C) 35(0.6)³ (0.4)⁴. D) (0.6)³ (0.4)⁴
*** A man takes step forward with a probability 0.6 and a step backward with a probability of 0.4. Find the probability that at the end of 11 steps, the man is:
91) one step ahead of starting point
A) (0.6)⁶ (0.4)⁵. B)462.(0.6)⁶ (0.4)⁵
C) 462(0.6)⁵ (0.4)⁶. D) 462(6/25)⁵
92) One step behind of starting point.
A) (0.6)⁶ (0.4)⁵. B) 462(0.6)⁶ (0.4)⁵
C) 462(0.6)⁵ (0.4)⁶. D) 462 (6/25)⁵
93) one step away of starting point
A) (0.6)⁶(0.4)⁵. B)462.(0.6)⁶ (0.4)⁵
C) 462(0.6)⁵ (0.4)⁶. D)462 (6/25)⁵
94) 8 coins are thrown simultaneously. the probability of getting at least 6 heads is:
A) 37/256 B) 73/256
C) 108/256. D) 64/256
95) what is the probability of guessing correctly at least six of the 10 answer in a true - false objective test
A) 139/512. B) 193/512
C) 391/624 D) none
96) A Binomial random variable X satisfies the relation 9P(x= 4)= P(x=2) when n= 6. find the value of the parameter P
A) 1/2 B)!1/3 C) 1/4 D) N
97) in a Binomial Distribution consisting of 5 independent trials, probability of 1 and 2 successes are 0.4096 and 0.2048 respectively. find the parameter P of the distribution ?
A) 1/3 B) 1/4 C) 2/5. D) 1/5
*** The incidence of occupational disease is such that on the average 25% of workers suffer from it. if 10 workers are selected at random, find the probability that:
98) Exactly two workers suffers from the disease.
A) 295245/1048576
B) 551124/1048576
C) 421155/1048576. D) none
99) not more than two workers suffer from the disease.
A) 295245/1048576
B) 551124/1048576
C) 421155/1048576. D) none
100) The overall percentage of failures in a certain examination is 40. what is the probability that out of a group of 6 candidates at least 4 passed the examination?
A) 353/3125. B) 1071/3125
C) 1701/3125. D) none
*** If the probability of defective bolts be 1/10, find the following for the binomial distribution of defective bolts in a total of 400.
101) the mean
A) 20 B) 30 C) 45 D) 40
102) the standard deviation
A) 3 B) 4 C) 5 D) 6
*** Five unbiased coins are tossed simultaneously at random. find the probability of getting
103) Exactly 2 tails
A) 5/16 B) 8/16 C) 13//16 D)14/16
104) at most two tails
A) 14/16 B)13/16 C) 8/16 D)7/16
105) at least 2 tails.
A) 8/16 B) 9/16 C) 12/16 D) 13/16
106) In a binomial distribution with six independent trials, the probability of 3 and 4 successes are found to be 0.245 7 and 0.0819 respectively. Find parameter P of the binomial.
A) 3/13. B) 4/13 C) 5/13 D) 6/13
107) a lot of electronic components is known to contain 20% defective item. 2 person A and H select 8 and 7 components from it. If the total number of defective components found by them is more than two, they reject the lot. Find the probability the lot will be accepted
A) 181/25 (4/5)¹¹. B) (4/5)¹³.
C) 181/25 (4/5)¹³. D) none
*** an experiment succeed twice as often as fails. what is the probability that the next five trials, there will be :
108) 3 successes
A) 77/243 B) 81/243
C) 80/243. D) none
109) at least three successes
A) 80/243 B) 192/243
C) 77/243. D) none
** An experiments succeeds twice as many times as it fails. find the chance that in 6 trials, there will be at least 5 successes.
A) 256/729 B) 625/729
C) 519/729 D) none
*** an unbiased cubic dice tossed 4 times. what is the probability of obtaining:
111) No six
A) 630/1296. B) 650/1296
C) 526/1296 D) none
112) at least 6
A) 67/1296. B) 76/1296
C) 625/1296 D) 645/1296
113) all odd numbers.
A) 1/4 B) 1/12 C) 15/16 D) 1/16
114) at least one even number
A) 1/4 B) 1/12 C) 15/16 D) 1/16
*** if a dice is thrown 6 tiimes, calculate the probability that:
115) a score of 3 or less occurs on exactly two throws.
A) 15/64 B) 32/66 C)16/64 d) n
116) A score of more than two occurs on exactly 3 throws.
A) 16/729 B)160/729 C)13/729 D)n
117) A score of 5 or less occurs at least once.
A) 46655/46666 B)46655//46656
C) 1/2. D) None
*** The probability that a college student will graduate is 0.4. determine the probability that out of 5 students.
119) one graduate
A) 0.08 B) 0.26 C) 0.92 D) 0.29
120) at least one will be graduate
A) 0.26 B) 0.08 C) 0.299 D) 0.92
121) The mean number of success of a B. D (p+q)ⁿ is 240 where p is the probability of success. The S. D is 12. Calculate the values of p,q,n 600, p=2/5,q=3/5
122) A certain brand of razor blades is sold in packets of 5. The following is the frequency distribution of 100 packets according to the number of faulty blades in them.
No of blades:0 1 2 3 4-more
No. packets : 80172 1 0
Find the number of faulty blades per packet. Assuming that the distribution is binomial, estimate the probability that a blade taken at random from any packet will be faulty. 0.24, 0.048
17) A coin is tossed 400 times and it shows heads 220 times. Discuss whether the coin is unbiased or not.
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