PROBABILITY
BASIC SIMPLE SUM.
Q.1If a pair of unbiased coins is tossed, obtain probability of occurrence of (i) both heads, (ii) exactly one head, (iii) at least one head.
Q.2Two fair dice are rolled simultaneously. Find the probabilities of the following events.
A:The sum of the two numbers is even.
B:The sum of the two numbers is at least eight.
C:The product of the two number is less than 10.
Q.3A committee of 3 persons is to be formed from 3 company secretaries, 4 economists and 1 chartered accountant. What is the probability that (i) each of the three professions is represented in the committee? (ii) the committee consists of the chartered accountant and at least one economist?
Q.4If a three digit number is formed out of 2,3,6,8,5 without repeating any digit, find probability that it is divisible by 5.
Q.5A fair coin is tossed 4 times. Find the probability that we will get two heads and two tails.
Q.6Six men in a company of 20 employees are graduates. If 3 men are selected from 20 at random, what is the probability that they are all graduates? What is the probability that at least one is a graduate?
Addition theorem
Q.7Two dice are thrown together. What is the probability that sum of the numbers appearing on the uppermost faces is divisible by 3 or 4?
Q.8The probability that a student will solve accountancy problem is and the probability that he will not solve statistics problem is If the probability that the student solve at least one problem is , what is the probability that he will solve both the problems?
Q.9Two dice are thrown together. What is the probability that sum of the numbers on uppermost faces of two dice is 5 or number on the second die is greater than the number on the first die?
Q.10A card is drawn from a pack of 52 cards. What is the probability that (i) the card is either red or black? (ii) the card is either heart or a queen.
Q.11A factory employs both graduate and nongraduate workers. The probability that a worker chosen at random is a graduade is 0.67, that the worker is married is 0.72 and that the worker is a married graduade is 0.5. Find the probability that a worker chosen at random is
(i)agraduade or married or both, (ii) a married nongraduate.
Q.12A fair die is thrown two times. What is the chance that
(i)product of the numbers on the uppermost face is 6.
(ii)sum of the numbers on the uppermost face is 8.
(iii)sum of the numbers on the uppermost face is at least 11.
(iv)die shows the same number in both the tosses.
Set theory
Q.13In a certain residential suburb, 60% of all households subscribe to the morning newspaper, 80% subscribe to the evening newspaper and 50% subscribe to both papers. If a household is selected at random, what is the probability that it subscribes to at least one of the two newspapers?
Q.14Les A and B be two mutually exclusive and exhaustive events defined on the sample space S. If 3P (A) = P(B), Find P(A B).
Q.15A and B are any two events on the sample space S. P(A) = P(B) = and P(A B) = . Find the value of the following:
(i)P(A B), (ii) P(A B’), (iii) P(A’ B), (iv) P(A’ B’ ), (v) P (A’ B’)
Q.16A software company is bidding for computer programs A and B. Probability that the company will get program A is , the probability that the company will get program B is and probability that company will get both the programs is . What is the probability that the company will get at least one program?
Q.17A survey of families in a certain city showed that 60% of the families are having a washing machine, 55% are having microwave oven, 40% of those who are having a washing machine, are having a microwave oven. If a family is selected at random, find the probability that it has neither a washing machine nor a microwave oven.
Q.18If P(A) = , P(B) = , P(A B) = , find the values of the following:
(i)P(A B), (ii) P(A B’ ), (iii) P(A’ B),
(iv)P(A’ B’ ), (v) P(A’ B’ ).
Q.19The probability that machines of a certain company require service in warrant period is 0.30, while the probability that the dryers of the same company require service in warranty period is 0.10. If a customer purchases both a machine and a dryer made by this company, what is the probability that (i) both machine and dryer need warranty service? (ii) neither machine nor dryer require warranty service? (Assume that machine and dryer work independently)
Q.20An urn contains 3 white and 5 black balls. Another urn contains 2 white and 7 black balls. One urn is selected at random and a ball is drawn from it at random. Find the probability that the ball drawn is white.
Q.21A card is drawn from a well shuffled pack of 52 playing cards. It is kept aside. Then a second card is drawn from the remaining 51 cards. Find the probability that both the cards are queens.
Q.22Probability that a student A can solve a certain problem is and that another student B can solve it is . If both try independently, What is the probability that
(i) the problem is solve? (ii) the problem is not solved?
Q.23Two cards are drawn at random from a pack of 52 playing cards. Find the probability that both are kings or both are queens.
Q.24In a certain building there are 9 flats, 3 on each floor and all are unoccupied. Two friends are allotted one flat each in this building, at random. Find the probability that (i) both of them get flats on the same floor, (ii) they get flats on the first floor, (iii) they get flats on different floors.
Q.25The probability that A can shoot a target is and the probability that B can shoot it is . If A and B shoot independently of each other, find the probability that
(i) the target is not shot at all,
(ii) exactly one of A and B shoot the target,
(iii) the target is shot.
Q.26Computer of brands A and B are to be sold. A salesman has 50% and 40% chances of finding customers for brands A and B respectively. The computers can be sold independently. Given that the salesman can sell at least one of the computers, what is the probability that computer of brand A has been sold?
Q.27A lot contains 12 items of which 4 are defective. Two items are drawn at random from the lot one after the other without replacement. Find the probability that both the items are non-defective.
Q.28A financial analyst has developed a scoring system for security bonds and he found that if the score from a bond is less than 40, there is a probability of 0.85 that it will default within next 5 years. Given that a randomly chosen bond currently has 25% chance of a score less than 40, what is the probability that a bond that defaults within next 5 years had a score greater than are equal to 40?
Q.29A card is drawn from a pack of 52 cards. What is the probability that it is a face card, given that it is a red card?
Q.30A pair of fair dice is rolled. If the sum of the numbers appeared is 8, find the probability that one die shows number 3.
Q.31A certain item is manufactured on two machines A1 and A2 in a factory. The machines A1and A2 produce respectively 60% and 40% of the total output of this item. It is known that 2% and 1% of the items produced by the machines A1 and A2 are defective. From a day’s production by the two machines, one item is chosen at random and it is found to be defective. What is the probability that it was produced by the machine A1?
Q.32If P(A) = , P(B) = , P(A B) = , find P (A|B) and P (B|A).
Q.33Two fair dice are thrown. Find the probability that the sum of the points is at least 10 given that it exceeds 7.
Q.34Out of 50 members of a club, 40 like tea, 20 like coffee and 15 like both tea and coffee. A member is selected at random and it is found that he does not like coffee. Find the conditional probability that he likes tea.
Q.35The are two urns, urn I contains 4 white and 2 black balls and urn II contains 2 white and 5 black balls. An urn is chosen at random and one ball is drawn from it. It is found to be white. What is the probability that it came from urn II?
Q.36There are two suppliers A and B who supply 55% and 45% of the total requirement of a certain component. Past experience shows that proportions of defective components supplied by them are 0.01 and 0.02 respectively. One component is taken at random and it is found to be defective. What is the probability that it was supplied by (i) A, (ii) B?
Q.37A random experiment results in an integer outcome between 1 and 10 (both inclusive). All numbers are equally likely. Let A = event that an odd number occurs and B = event that a number divisible by 3 occurs. Obtain
(i)P(A|B), (ii) P(B|A), (iii) P(A’ | B), (iv) P(A|B’), (v) (A’ |B’).
Q.38The personnel department of a company has 100 engineers whose distribution is as given below:
Age (in years) / B.E. / M.E. / Total20 – 30 / 20 / 5 / 25
30 – 40 / 25 / 10 / 35
40 / 10 / 30 / 40
Total / 55 / 45 / 100
If one engineer is selected at random, find the probability that (i) he is only B.E., (ii) he is M.E. given that his age is 40, (iii) he is under 30 given that he has M.E. degree.
Space
Q.39Write down the sample space for each of the following random experiments:
(i)Two cards are drawn in succession from a pack of 52 playing cards and their colours are noted.
(ii)Five coins are tossed and number of heads obtained is noted.
(iii)From five persons A, B, C, D, E two persons are selected randomly to form a committee.
Q.40Three cards are drawn in succession from a pack of 52 playing cards and their coours are noted.
(i)Write down the sample space for this experiment.
(ii)Write down the subsets corresponding to the following events:
(a)Exactly two red cards are obtained.
(b)at least two red cards are obtained.
(c)at most two red cards are obtained.
(d)at most one red card is obtained.
Q.41Write down the sample space S for the random experiment of rolling two dice.
Write down the subsets of S defining the following events:
(i)both the dice show the same number.
(ii)die I shows an odd number and die II shows an even number.
(iii)die I shows 6.
(iv)the sum of the numbers on the two dice is at least 9.
Q.42State whether the following statements are true or false:
(i)Any event A and its complementary event Ac are mutually exclusive and exhaustive events.
(ii)Sample space for a random experiment is unique.
(iii)The impossible event is the complementary event of the certain event.
Set
Q.43In a survey conducted by a Music Club, it was observed that, 45% people out of 1000 liked Indian classical music, while 50% liked western music and 15% liked neither Indian nor Western music. How many individuals liked both the types of music?
Q.44100 students appeared for two examinations. 60 passed in the first examination, 50 passed the second and 30 passed in both. Find the probability that a student selected at random
(i)Passed in at least one examination.
(ii)Passed in exactly one examination.
(iii)Failed in both the examinations.
Q.45Events A, B, C from a partition of sample space S. If 3 P (A) = 2 P(B) = P(C), find P(AB).
Basic
Q.46An unbiased coin is tossed 4 times. Find the probability that it shows head exactly two times.
Q.47Nine digits 1,2,3,……. 8,9 are arranged in a row to form a nine digit number. Find the probability that the digits 4,5,6 are together (i) in the order 654, (ii) in any order.
Add
Q.48Two dice are thrown together. What is the probability that sum of the numbers on two dice is 5 or number on the second die is greater than or equal to the number on the first die.
Q.49If P(A) = P(B) = and P(AB) = , find the values of the following:
(i)P (A B),(ii)P(A B’), (iii) P(A’ B)
Q.50The probability that a contractor will get a plumbing contract is 0.4 and that he will not get an electric contract is 0.7. If the probability of getting at least one contract is 0.6, what is the probability that he will get (i) both the contracts? (ii) exactly one contract?