Probability And Statistics *********************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. 5 boys and 4 girls ii. 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 are defective. i. What is the probability that a car purchased is defective? ii. 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)= when X≥0,find i. k ii. Mean iii. Variance. *********************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
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 www.jntuworld.comwww.jntuworld.com JNTUWORLD F(x)= 3 0≤ x <1 = 0 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. X01234567 F 0 k 2k 2k 3k 7 +k i. K ii. Mean iii. Variance iv. Smallest value of x such that P(X≤x)>1/2. 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 www.jntuworld.comwww.jntuworld.com JNTUWORLD = -kx + 6k if 4≤ x <6 Find k and mean value of X. 15. For the discrete probability distribution. x01234567 f 0 K 2k 2k 3k +k
Determine i. K ii. Mean iii. Variance. 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)= when x≥0,find iv. K v. Mean vi. 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
iii. 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. x012345 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 x012345 f 38 144 342 287 164 25 14. Fit the poison distribution to calculate the theoretical frequencies for the following data. x01234 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. x01234 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
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 i. P(x≥1) ii. P(x≤3) iii. P(2≤ x ≤5). 42. 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/-. 43. 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. 44. 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). 45. Using the recurrence formula find the probabilities when x=0, 1,2,3,4 and 5. If the mean of poison distribution is 3. 46. 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, 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. 6. 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. 7. Let S={1,5,6,8}. Find the probability distribution of the sample mean for random sample of size 2 drawn without replacement. 8. 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. 9. Define the statistics t and F and write down their sampling distributions. State the important assumptions in respect of them.
10. 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. 11. 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). 12. 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, ). Find the probability that the ratio of the sample variances does not exceed 2.28.
