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PROF. BHAMBWANI’S RELIABLE CLASSES STATISTICS WORD FILE PROBABILITY DISTRIBUTION LEC 1: THEORY LEC 2: THEORY LEC 3: BINOMIAL DISTRIBUTION Q1. A fair coin is tossed 6 times. Find the probability of getting (a) 3 heads (b) at least 4 heads. (Ans.: 5/16, 11/32) Q2. A dice is tossed 5 times. What is the probability that 6 shows up exactly thrice. Ans.: (0.032) (N 84) Q3. A’s chance of winning a single game against B is 2/5. Find the chance that in a series of 5 games, A wins. (a) Exactly three games, (b) Atleast three games. (Ans.: 144/625,992/3125) Q4. 60% of T.V. viewers watch ‘Chhaya Geet’. If 5 T.V. viewers are selected at random, what is the probability that 4 of them watch ‘Chhaya Geet’? (Ans.: 162/625) Q5. A has won 20 out of 30 games of chess with B. In a series of 6 games. What is the probability that A would win 4 or more games? (Ans.:496/729) Q6. Probability that a bomb dropped from a plane hits a target is 0.4. Two bombs can destroy a bridge. If in all 6 bomb are dropped, find the probability that the bridge will be destroyed. (Ans.: 0.76672) Q7. A fair coin is tossed 10 times. Find the probability of getting atleast 7 heads. (Ans.: 11/64) LEC 4: Q8. A person fires 5 shots at a target. The probability that a shot will hit the target is 4/5. Find the probability that the target is hit (i) exactly 2 times. (ii) atleast once. (iii) all 5 times. (Ans.: 32/625, 3124/3125, 1024/3125) Q9. 20% of the workers in a factory are females. Six workers are selected at random on a committee. Find the probability that there are atleast two female workers on the committee. (Ans.:1077/3125) Q10. If the chance that any one of the 5 telephone lines is busy at any instant is 0.1, find the probability that (i) all the lines are busy. (ii) not more than 3 lines are busy. (Ans.: 0.00001, 0.99954) Q11. Four coins are tossed simultaneously. what is the probability of getting (i) 2 heads and 2 tails, (ii) at least two heads, and iii) at least one head. Q12. If a dice is rolled 3 times, what is the probability of 5 coming up at least once? (Ans. 0.4213) (N 83) Q13. The incidence of occupational disease in an industry is such that the workmen have a 20% chance of suffering from it. What is the probability that out of six workmen, 4 or more will contract the disease? Ans. 0.0169 Q14. If we take 1280 sets each of 10 tosses of a fair coin, in how many sets should we expect to get 7 heads and 3 tails? (Ans.150. (N 87)

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Page 1: reliable.redik.inreliable.redik.in/.../PROBABILITY_DISTRIBUTION.docx  · Web viewQ5.A has won 20 out of 30 games of chess with B. ... Q9.20% of the workers in a factory are females

PROF. BHAMBWANI’SRELIABLE CLASSES

STATISTICS WORD FILEPROBABILITY DISTRIBUTION

LEC 1:THEORY

LEC 2:THEORY

LEC 3:BINOMIAL DISTRIBUTION

Q1. A fair coin is tossed 6 times. Find the probability of getting (a) 3 heads (b) at least 4 heads. (Ans.: 5/16, 11/32)

Q2. A dice is tossed 5 times. What is the probability that 6 shows up exactly thrice. Ans.: (0.032) (N 84)

Q3. A’s chance of winning a single game against B is 2/5. Find the chance that in a series of 5 games, A wins. (a) Exactly three games, (b) Atleast three games. (Ans.: 144/625,992/3125)

Q4. 60% of T.V. viewers watch ‘Chhaya Geet’. If 5 T.V. viewers are selected at random, what is the probability that 4 of them watch ‘Chhaya Geet’?(Ans.: 162/625)

Q5. A has won 20 out of 30 games of chess with B. In a series of 6 games. What is the probability that A would win 4 or more games? (Ans.:496/729)

Q6. Probability that a bomb dropped from a plane hits a target is 0.4. Two bombs can destroy a bridge. If in all 6 bomb are dropped, find the probability that the bridge will be destroyed. (Ans.: 0.76672)

Q7. A fair coin is tossed 10 times. Find the probability of getting atleast 7 heads. (Ans.: 11/64)

LEC 4:Q8. A person fires 5 shots at a target. The probability that a shot will hit the target

is 4/5. Find the probability that the target is hit (i) exactly 2 times. (ii) atleast once. (iii) all 5 times. (Ans.: 32/625, 3124/3125, 1024/3125)

Q9. 20% of the workers in a factory are females. Six workers are selected at random on a committee. Find the probability that there are atleast two female workers on the committee. (Ans.:1077/3125)

Q10. If the chance that any one of the 5 telephone lines is busy at any instant is 0.1, find the probability that (i) all the lines are busy. (ii) not more than 3 lines are busy. (Ans.: 0.00001, 0.99954)

Q11. Four coins are tossed simultaneously. what is the probability of getting (i) 2 heads and 2 tails, (ii) at least two heads, and iii) at least one head.

Q12. If a dice is rolled 3 times, what is the probability of 5 coming up at least once? (Ans. 0.4213) (N 83)

Q13. The incidence of occupational disease in an industry is such that the workmen have a 20% chance of suffering from it. What is the probability that out of six workmen, 4 or more will contract the disease? Ans. 0.0169

Q14. If we take 1280 sets each of 10 tosses of a fair coin, in how many sets should we expect to get 7 heads and 3 tails? (Ans.150. (N 87)

Q15. 8 coins are tossed at a time, 256 times. Find the expected frequencies of success (getting a head) and tabulate the result obtained. Find the mean and S.D. of the fitted distribution.

Q16. Five coins are tossed 3,200 times, find the frequencies of the distribution of heads and tails and tabulate the results. Calculate the mean number of success and standard deviations. (C.A. May 1977)

Q17. How many tosses of a coin are needed so that the probability of getting at least one head is 87.5% (Ans. 3) (N 84)

Q18. (a) Comment on the following:For a binomial distribution, mean =7, variance = 11.

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Q19. For a binomial distribution, the mean is 6 and the standard deviation is 2. Write out all the terms of the binomial distribution.

Q20. Find the Binomial distribution whose mean is 9 and variance being 2.25

LEC 5:Q21. Fit a binomial distribution to the following data:

X : 0 1 2 3 4 5Y : 10 10 30 25 15 10 (M-02)

Q22. The mean of a binomial distribution is 20 and standard deviation is 4. Calculate n, p and q with usual notations. (N-03)

Q23. Determine the binomial distribution for which the means is 4 and standard deviation is 3.

Q24. Point out the fallacy if any in the following statement The mean of a binomial distribution is 10 and its standard deviation is 4.

Q25. State the conditions under which a binomial distribution is used. Find the binomial distribution if the mean is 12 and the standard deviation is 2.

(May 1997)Q26. In a Binomial distribution with 6 independent trials, the probabilities of 3 and 4

successes are found to be 0.2457 and 0.0819 respectively. Find the parameter ‘p’ of the Binomial. (June 93)

Q27. A binomial distribution consisting of five independent trials, probabilities of 1 and two success are 0.4096 and 0.2048, respectively. Find the parameter `p` of the distribution

Q28. A binomial random variable X satisfies the relation 9P(X =4) = P (X = 2) when n = 6. Find the value of Parameter `P'.

LEC 6:THEORY ON POISSON DISTRIBUTION Q1. A manufacturing concern employing a large number of workers finds that, over a

period of time, the average absentee rate is three workers per shift. Calculate the probability that, on a given shift,

i) exactly two workers will be absent,

ii) more than four workers will be absent.

Q2. Between 2 and 4 p.m. the average number of phone calls per minute coming into the switchboard of a company is 2.5 Find the probability that during one particular minute there will be (i) no phone call at all, ii) exactly 3 calls, iii) at least 7 calls.

LEC 7:Q3. The average number of customers, who appear at a counter of a certain bank per

minute, is two. Find the probability that during a given minute (i) no customer appears, (ii) three or more customers appear. (N-03)

Q4. Using Poisson approximation to the Binomial distribution, solve the following problem :If the probability that an individual suffer a bad reaction from a particular injection is 0.001, determine the probability that out of 2,000 individuals i) exactly three, ii) more than two, individuals will suffer a bad reaction

LEC 8:Q5. In a town 10 accidents take place in a span of 50 days. Assuming that the number of

accidents follows the Poisson distribution, find the probability that there will be 3 or more accidents in a day.

Q6. A company knows on the basis of past experience that 3% of its bulbs are found to be defective. We have to find out the probability of 0, 1,2, 3, 4 and 5 defectives in a sample of 100 items

LEC 9:Q7. The number of accidents in a year attributed to taxi drivers in a city follows

Poisson distribution, with mean 3. Out of 1,000 taxi drivers, find approximately the number of drivers with (i) no accident in a year and ii) more than 3 accidents in a year. (Given : e-1 = 0.3679, e-2=0.1353, e-3= 0.0498) (May 04)

Q8. Records show that the probability is 0.000102 that a car will have a flat tyre while driving over certain bridge. Use the Poisson distribution find the probability that among 20,000 cars driven over this bridge, not more than one will have a flat tyre. (Given e = PT-1 PT = 0.89, e-0.4=0.670) Ans. a) 0.9380 (M 90)

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Q9. Fit a Poisson distribution to the following data and calculate the theoretical frequencies:Number of mistakes per page: 0 1 2 3 4No. of pages on which mistakes occurred. 109 65 22 3 1e-0.61 =0.5434

LEC 10:Q1. Given a normal distribution with X = 50 and σ = 8, find the probability that X

assumes a value between 34 and 62. _

Q2. Given a normal distribution with X = 200 and σ = 25, find the probability that X assumes a value greater than 240.

Q3. The customer accounts of a certain departmental store have an average balance of Rs 120 and a standard deviation of Rs 40. Assuming that the account balances are normally distributed:-i) What proportion of the account is over Rs 150?ii) What proportion of the accounts is between Rs 100 and Rs150?iii) What proportion of the accounts is between Rs 60 and Rs 90?

Q4. In a sample of 120 workers in a factory the mean and standard deviation of wages were Rs 11.35 and Rs 3.03 respectively. Find the percentage of workers getting wages between Rs 9 and Rs 17 in the whole factory assuming that the wages are normally distributed.

Q5. A Sales tax officer has reported the average sales of the 500 firms that the has to deal with during a year amount to Rs 72,000 with a standard deviation of Rs 20,000. Assuming that the sales in these firms are normally distributed, find -

i) The number of firms whose sales are over Rs 80,000 andii) the percentage of firms whose sales are likely to range between Rs 60,000 and Rs

80,000.

Q6. A grinding machine is set so that its production of shafts has an average diameter of 10.10 cms and a standard deviation of 0.20 cms. The product specification call for shaft diameters between 10.05 cms and 10.20 cms. What proportion of output meets the specifications presuming normal distribution? Ans.29.02% (N 85)

Q7. In a sample of 1,000 items, the mean weight is 45 kg with a standard deviation of 15 kg. Assuming the normality of the distribution, find the number of items weighing between 40 and 60 kgs. Ans. 471

Q8. The marks obtained in a certain examination follow normal distribution with mean 45 and standard 10. If 1000 students appeared at the examination. Calculate the number of students scoring (i)less than 40 marks and (ii) more than 60 marks. (May 1999)

Q9. In a sample of 240 workers in a factory, the mean and standard deviation of wages were Rs. 113.50 and Rs. 30.30 respectively. Find the percentage of workers getting wages between Rs. 90 and Rs. 170 in the whole factory assuming that the wages are normally distributed. (N-2002)

Q10. A random variable X is normally distributed with mean = 12 and s.d. = 2. Find P(9.6<X<13.8). Given that the area:

A = 0.3159 for Z = 0.9And A = 0.3849 for Z = 1.2Where Z is a standard normal variety. (N-2000)

LEC 11:Q11. The income distribution of officers of a certain company was found to follow normal

distribution. The average income of an officer was Rs. 15,000.00. the Standard deviation of the income of officers was Rs.5,000.00. If there were 242 officers drawings salary above Rs.18,500.00, how many officers were there in the company?

(N-01)

Q12. In a sample of 1,000 items, the mean weight and standard deviation are 50 and 10 kilograms, respectively. Assuming the distribution to be normal, find the number of items weighing between 40 and 70 kilograms. (N-04)

Q13. Of a large group of men, 5% are under 60 inches in height and 40% are between 60 and 65 inches. Assuming a normal distribution, find the mean and height and standard deviation. _

Q14. Given a normal distribution with X = 50 and σ = 10, find the values of X that has (a) 13% of the area to its left and (b) 14% of the area to its right.

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Q15. Assume the mean height of students in an exactly normal distribution to 68.22 inches with a variance 10.8 inches. How many students in a college of 1,000 students would you expect to be over six feet tall? (area under normal curve corresponding to z = 1.15 is 0.3749) Ans. 125 (M 85)

Q16. In a large group of men, it is found that 5% are under 60 inches and 40% are between

60 and 65 inches in height. Assuming the distribution to be exactly normal, find the mean and the standard deviation of height. ( For areas of the standard normal curve 0.05 and 0.45 between 0 and specified vales, the z values are approx. 0.125 and 1.645 respectively). (Ans. x = 65.41,y = 3.29)

(N 92)

Q17. In distribution exactly normal 9.85% of the items are under 40 and 89.97% are under 60. What are the mean and standard deviation of the distribution? Ans. x= 50.039, 0= 7.782 (N 92)

LEC 12:Q18. Let X be a continuous variable and follows a normal distribution with mean 12 and

standard deviation 2. What is the probability that the value of X selected at random lies in the interval (11,14)? (Nov.1997)

Q19. The incomes of a group of 10,000 persons were found to be normally distributed with mean equal to Rs.750 and standard deviation equal to Rs.50. What was lowest income among the richest 250? (J-94)

Q20. A random variable X has the following probability distribution:Value of X : -2 -1 0 1 2 3P(X = x) : 0.1 K 0.2 2K 0.3 K(ii) Find the value of K.(ii) Find the expected value and variance of X. (N-99)

PROBLEMS ON EXPECTATION (MEAN) AND VARIANCE OF PROBABLITY DISTRIBUTIONQ21. The probability distribution of a random variable X is given as:

X = -2, 3, 1 P(X) = 1/3, 1/2, 1/6

Find E(2X + 5) and E(X2). (M-2000)

Q22. Define mathematical expectation. For the following probability distribution, calculate the mean and the variance of the random variable x:X : 8 12 16 20 24P(X) : 1 1 3 1 1

8 6 8 4 12 (M-01)

LEC 13:THEORY ON THEORTICAL DISTRIBUTION

LEC 14:THEORY

LEC 15:

Model paper Objectives

Paper -11) For Poisson Distribution, a) Mean and standard deviation are equal b) Mean and variance are equal c) Standard deviation and variance are equal d) Both (a) and (b)

2) ________ distribution is symmetrical when p = 0.5 a) chi – square b) Population c) Poisson d) Binomial

3) The quartile deviation of a normal distribution with mean 10 and standard deviation 4 is a) 0.675 b) 67.50 c) 2.70 d) 3.20

4) The mean of a binomial distribution with parameter n and p is a) n(1−p) b) np (1−p) c) np d) √np(1−p)

5) The total area of the normal curve is a) One b) 50 per cent c) 0.50 d) Any value between 0 and 1

6) The method usually applied for fitting a binomial distribution is known as ________. a) Method of least square b) Method of moments c) Method of probability d) Method of deviations

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7) Poisson distribution may be a) Unimodal b) Bimodal c) Multi – modal d) (a) or (b)

8) If the quartile deviation of a normal curve is 4. 05, then its mean deviation is a) 5.26 b) 6.24 c) 4.24 d) 4.80

9) A binomial distribution is a) Never symmetrical b) Never positively skewed c) Never negatively skewed d) Symmetrical when p = 0.5

10) ________ distribution is sometimes known as the “distribution of rare events”.a) Poisson b) Normal c) Binomial d) t

11) The Standard Deviation of the ________ distribution is called standard errora) Normal b) Poisson c) Binomial d) Sampling

12) In binomial distribution if n = 4 and p = 13 , then the value of variance is

a) 83 b)

89 c)

43 d) None of these

13) Standard deviation of binomial distribution is a) (npq)2 b) √npq c) (np)2 d) √np

14) The normal curve is _______a) Bell - shaped b) u - shaped c) J – shaped d) Inverted J - shaped

15) The mean of binomial distribution is 4 and standard deviation √3. What is the value of n? a) 14 b) 16 c) 18 d) 20

16) The result of ODI matches between India and Pakistan followsa) Binomial distribution b) Poisson distribution c) Normal distribution d) (b) or (c)

17) If in a binomial distribution mean = 20, Standard Deviation = 4 then n is equal to a) 80 b) 100 c) 90 d) None of these

18) In binomial distribution ‘p’ denotes probability of __________ a) Success b) Failure c) Both of these d) None of these

19) If in a binomial distribution Mean = 20, Standard deviation = 4 then p is equal to

a) 25 b)

35 c)

15 d)

45

20) The normal curve is a) Positively skewed b) Negatively skewed c) Symmetrical d) All of these

LEC 16:

Paper 21) Variance of a binomial distribution is always ________ its mean.

a) Equal to b) More than c) Less than d) None of these

2) If in binomial distribution mean is 10 and S.D. is 2, q will be __________.a) 0 b) 0.2 c) 0.8 d) 0.4

3) If in binomial distribution np=6 and npq=3, then p is equal to

a)13 b)

12 c)

14 d) None of these

4) We use T – distribution when samples are drawn from the _________ population a)Normal b) Binomial c) Poisson d) None of these

5) __________ is uniparametric distribution a) Binomial b) Poisson c) Normal d) Hyper geometric

6) What is the number of trials of a binomial distribution having mean and Standard Deviation as 3 and 1.5 respectively?a) 2 b) 4 c) 8 d) 12

7) The mean of binomial distribution is 4 and standard deviationis√3. What is the value of p?

a) 13 b)

14 c)

15 d)

34

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8) A _______ distribution is a theoretical distribution that expresses the functional relation between each of the distinct values of the samples statistic and the corresponding probability. a) Normal b) Binomial c) Poisson d) Sampling

9) In ________distribution mean = variance.a) Normal b) Binomial c) Poisson d) None of above

10) When ‘p’ = 0.5, the binomial distribution is a) Asymmetrical b) Symmetrical c) Both of above d) None of above

11) Standard deviation of poisson distribution is

a) m b) m2 c) √m d) 1

√m

12) The mean of binomial distribution is a) (np )2 b) np c) √np d) None of these

13) If the mean deviation of a normal variable is 16, what is its quartile deviation? a)10.00 b) 13.50 c) 15.00 d) 12.05

14) The most important continuous probability distribution is know is (a) Binomial distribution (b) Normal distribution(c) Chi-Square Distribution (d) Sampling Distribution

15) A binomial distribution is (a) Never Symmetrical (b) Never Negatively skewed (c) Symmetrical when p = 0.5 (d) never positively skewed

16) The result of ODI matched between INDIA and ENGLAND follows : (a) Binomial distribution (b) Poisson distribution (c) Normal distribution (d) (b) or (c)

17) The mode and mean of a normal distribution (a) May be equal (b) May be different (c) Are always equal (d) (a) or (b)

18) The mean of binomial distribution with parameters n and p is ________

(a) n(1-p)(b) np(1-p) (c) np (d) √np(1−p )

19) In normal distribution, as the distance from __________ decreases, the curve goes far and far to the horizontal axis. (a) Median (b) Mode (c) Mean (d) None of these

20) For a binomial distribution mean and mode are not equal, when q = 0.50 (a) True (b) False (c) Both (d) None of these