PROF. BHAMBWANI’S
RELIABLE CLASSES
STATISTICS WORD FILE
PROBABILITY
LEC 1:
THEORY
LEC 2:
Unit I -Problems on Simple Events
Q1.Write the sample space for the event that 3 coins are
tossed.
Q2.Two fair coins are tossed at a time Write the sample space of this random experiment and state the number of points in the sample space.
Q3.A perfect cubic die is thrown. Find the probability that an even number comes up.
Q4.A perfect cubic die is thrown. Find the probability that a perfect square comes up.
Q5.If n(S) = 36, P(A) = 0.5 Find n(A)
Q6.A card is drawn from a well shuffled pack of 52 playing cards.
Find the probability that is not a face card
Q7.A card is drawn from a well shuffled pack.
Find the probability of getting a face card.
Q8.Two unbiased dice are thrown. Find the probability that the sum of the scores on their upper faces is 8.
Q9.A fair die is thrown. Find the probability of getting a number on the upper face which is (i) even (ii) greater than 2.
Q10.A bowl contains 100 slips numbered 1 to 100. A slip is drawn at random from the bowl. Find the probability that the slip bears a number which is divisible by 5.
Q11.A bowl contains 100 slips numbered 1 to 100. A slip is drawn at random from the bowl. Find the probability that the slip bears a number which is divisible by 7.
Q12. Two dices are thrown what is the probability that the sum of the scores on the uppermost faces is i) Seven ii) Six
Q13.Given below are the weekly wages (in Rupees) of six workers in a factory:
62 90 78 85 79 68
If two of these workers are selected at random to serve as representatives, what is the probability that at least one will have a wage lower than the average?
LEC 3:
THEORY ON ADDITION THEOREM
LEC 4:
Q16That is the chance of throwing at least 7 in a single cast with 2 dice?
(a)5/12(b)7/12(c)1/4(d)17/36
Q17.A, B and C are three mutually exclusive and exhaustive events such that P (A) = 2 P (B) = 3P(C). What is P (B)?
(a)6/11(b)6/22(c)1/6(d)1/3
Q18.A bag contains 12 balls which are numbered from 1 to 12. If a ball is selected at random, what is the probability that the number of the ball will be a multiple of 5 or 6?
(a)0.30(b)0.25(c)0.20(d)1/3
Q19.If two unbiased dice are rolled, what is the probability of getting points neither 6 nor 9?
(a)0.25(b)0.50(c)0.75(d)0.80
LEC 5:
THEORY ON MULTIPLICATION THEOREM
LEC 6:
Q20.One lottery ticket is drawn at random from a set of 25 tickets numbered 1 to 25. Find the probability that the number on the ticket drawn is
(i)either odd or square of an integer
(ii)multiple of 4 or 5 (iii) even number or divisible by 5
[(i)3/5 (ii)2/5 (iii)3/5]
Q21.The probability that a man will be alive for 60 years is 3/5 and that his wife will be alive for 60 years is 2/3 Find the probability that,
i)Both will be alive for 60 years.
ii)Only the man will be alive for 60 years.
iii)None will be alive for 60 years.
Q22.Two cards are drawn at random from a pack of 52 playing cards. Find the number of points in the sample space for this random experiment.
Q23.Two cards are drawn from pack of 52 well shuffled cards. Find the probability that FIRST is a king and the SECOND is a queen.
Q24.Two cards are drawn from a pack of well shuffled cards find the
probability that one is a spade and the other is an ace.
Q25.From a pack of 52 cards, two cards are drawn at random. Find the probability that both are spade cards.
Q26.Two Cards are drawn from a pack of 52 playing cards. One by one without replacement. Find the probability that the
I)first card drawn is a king and second is not a king.
II)One is king and other is queen.
Q27.A bag contains, 7 blue balls and 10 yellow balls, 2 balls are drawn at random. Find the probability that both are of different colour.
Q28.The probability that A can solve a problem is 1/2 and that B can solve the same problem is 2/3, if both of them try independently, find the probability that the problem is solved.
Q29.A bag contains 6 white and 9 black balls. If three balls are drawn at random find the probability that all are black.
Q30.Two cards are drawn at random from a pack of 52 cards. Find the probability that they are of different suits.
LEC 7:
Q31.The chance of A winning a race is 1/6 and chance of B winning a race is 1/8. What is the chance that neither of the two should win?
Q32.Two cards are drawn one after the other from a pack of 52 cards. Find the probability that both the cards are kings, when
(i)The first card is replaced.
(ii)The first card is not replaced.
Q33.The probability that a student ‘A’ can solve a problem is 1/3 ‘B’ can solve it is 1/2 and ‘C’ can solve it is 1/4, if all of them try independently, what is the probability that the problem is solved.
Q34.A problem is given to three students whose chances of solving it are 1/2, 1/3, 1/4 respectively. Find the probability that the problem will not be solved considering that they are trying independently.
Q35.A purse contains 4 silver coins and 5 copper coins. Another purse contains 3 silver and 4 copper coins. A purse is selected at random and a coin is drawn from it at random. What is the probability that it is a copper coin?
Q36.An urn contains 3 white and 5 red balls and another urn contains 2 white and 4 red balls. One urn is selected at random and a ball is drawn from it at random. Find the probability that the ball is red.
Q37.An urn contains 3 red and 4 black balls and another urn contains 2 red and 4 black balls. One urn is selected at random and a ball is drawn from it at random. Find the probability that the ball is red.
Q38.Three children Seeta, Geeta and Raju often help their parents in the kitchen. The respective probability of their breaking a dish in the kitchen are 1/3 , 1/2 , and 4/5, if one of them is selected at random to help with the dishes. Find the probability that a dish broken.
Q39.Two students appear for an examination.Their chances of passing the examination being 0.7 and 0.8 respectively. Find the probability that
atleast one of them passes the examination.
LEC 8:
SAME AS LEC 7 + THEORY ON COMBINATION
LEC 9:
Q40.A box contains 10 radio valves of which 4 are defective. Find the probability that if two valves are taken from the box, they are both defective.
Q41.A bag contain 6 blue,4 white and 5 purple marbles. Three marbles are taken out at random, What is the probability that i)all the three are blue. ii)the three are of three different colours.
Q42.A box contains 5 blue,6 black and 8 green marbles are drawn at random what is the probability that i)two are blue and 1 black ii)two are green and 1 black iii)all the three are black. iv) One of each colour.
Q43.A housewife buys a dozen eggs of which two are bad. She chooses 4 eggs to scramble for breakfast. Find the probability that she chooses i)All good eggs; ii)three good and 1 bad egg; iii)2 good and 2 bad eggs; iv)at least 1 bad egg.
Q44.From a group of 5 men and 4 women, 4 persons are selected at random to form a committee. What is the probability that the committee contains 3 men and a woman?
Q45.From a well-shuffled pack of 52 cards, 3 cards are drawn at random. Find the probability that three cards contain two kings and one ace.
Q46.A room has 3 electric lamps. From a selection of 12 electric bulbs of which 8 are good, 3 bulbs are selected at random and put in the sockets. Find the room is lighted by at least one of bulbs.
Q47.An urn contains 5 blue and an unknown number x of red balls. When two balls are drawn at random the probability of both of them being blue is 5/14 find x.
LEC 10:
Q55.A box contains 5 white and 7 blacks balls. Two successive drawn of 3 balls are made (i) with replacement (ii) without replacement. The probability that the first draw would produce white balls and the second draw would produce black balls are respectively.
(a)6/321and 3/926(b)1/20 and 1/30
(c)35/144 and 35/108(d)7/968 and 5/264
Q56.There are three boxes with the following composition:
Box I: 5 Red + 7 White + 6 Blue balls
Box II: 4 Red + 8 White + 6 Blue balls
Box III: 3 Red + 4 White + 2 Blue balls
If one ball is drawn at random, then that is the probability that they would be of same colour?
(a)89/727(b)97/729(c)82/729(d)23/32
Q57.A bag contains 8 red and 5 white balls. Two successive draws of 3 balls are made without replacement. The probability that the first draw will produce 3 white balls and the second 3 red balls is
(a)5/223(b)6/257(c)7/429(d)3/548
Q58.There are two boxes containing 5 white and 6 blue balls and 3 white and 7 blue balls respectively. If one of the boxes is selected at random and a ball is drawn from it, then the probability that the ball is blue is
(a)115/227(b)83/250(c)137/220(d)127/250
Q59.A problem in probability was given to three CA students A, B and C whose chance of solving it are 1/3, 1/5 and 1/2 respectively. What is the probability that the problem would be solved?
(a)4/15(b)7/8(c)8/15(d)11/15
Q60.There are three persons aged 60,65 and 70 years old. The survivals probabilities for these three persons for another 5 years are 0.7, 0.4 and 0.2 respectively. What is the probability that at least two of them would survive another five years?
(a)0.425(b)0.456(c)0.392(d)0.388
Q61.Tom speaks truth in 30 percent cases and Dick speaks truth in 25 percent cases. What is the probability that they would contradict each other?
(a)0.325(b)0.400(c)0.925(d)0.075
Q62.There are two urns. The first urn contains 3 red and 5 white balls whereas the second urn contains 4 red and 6 white balls. A ball is taken at random from the first urn and is transferred to the second urn. Now another ball is selected at random from the second arm. The probability that the second ball would be red is
(a)7/20(b)35/88(c)17/52(d)3/20
Q63.A packet of 10 electronic components is known to include 2 defectives. If a sample of 4 components is selected at random from the packet, what is the probability that the sample does not contain more than 1 defective?
(a)1/3(b)2/3(c)13/15(d)3/15
Q64.8 identical balls are placed at random in three bags. What is the probability that the first bag will contain 3 balls?
(a)0.2731(b)0.3256(c)0.1924(d)0.3443
Q65.X and Y stand in a line with 6 other people. What is the probability that there are 3 persons between them?
(a)1/5(b)1/6(c)1/7(d)1/3
LEC 11:
SET THEORY
LEC 12:
THEORY ON CONDITIONAL PROBABILITY
LEC 13:
Q68. Given:- P(A B) = 0.25, n(A B) = 13, Find n(S).
Q69. If A and B are two events in a sample space S such that P(A) = 0.8, P(B) =
0.6 and P(A n B) = 0.5 Find P(A U B) and P(A/B)
Q74.The department of a company has records which show the following analysis of its 200 Members
Age Bachelor’s Master’s Total
Degree Degree
Under 30 90 10 100
30 to 40 20 30 50
over 40 40 10 50
Total 150 50 200
If one engineer is selected at random from the company find
a) The probability he has only a bachelor’s degree
b) The Probability he has a master’s Degree given that he is over 40
c) The probability he is under 30 given that he has only a bachelor’s degree.
Q75.Calculate Pr. (B\A) if (A) = 0.75 Pr. (B) = 0.60 and Pr (A\B)=0.90
Q76.A and B are two events such that P(A) = 0.8, P(B) = 0.6,
P(A B)=0.5.
Find (i)P(A U B) (ii) P(A/B) (iii) P(B/A)
Q77.P(A) = 0.3, P(B)=0.4 and P(A/B)= 0.32. Find (i)P(A B) (ii)P(B/A)
Q78.A and B are two events such that P(A) = 2/3, P((B’) = 3/4, P(A/B)= 4/5, Find P(A B) and P(B/A)
Q79.It is known that 20 % of the males and 5% of the females are unemployed in a certain town consisting of an equal number of males and females. A person is selected at random and is found to be unemployed What is the probability that he is i) a male ii) a female.
Q80. There are 100 students in a college class of which 36 are boys studying Statistics and 13 girls are not studying statistics. If there are 55 girls in all, find the probability that a boy picked up at random is not studying Statistics.
Q81. In an examination, 30% of the students have failed in Mathematics, 20% of the students have failed in Chemistry and 10% have failed in both Mathematics and Chemistry. A student is selected at random.
i) What is the probability that the student has failed in Mathematics if it known that he has failed in Chemistry?
ii) What is the probability that the student has failed either in Mathematics or in Chemistry?
Q82.A bag contains 3 red and 2 white balls. A second bag contains 2 red and
4 white balls. One ball is selected at random from the first bag and transferred to the second bag. Then a ball is drawn at random from the second bag. Find the probability that it is a red ball. Ans:13/35
Q83.A purse contains 4 five rupee coins and 3 ten rupee coins. Another purse
contains 2 five rupee coins and 4 ten rupee coins. If a coin is selected
at random from one of the two purses, find the probability that it is a
five rupee coin.
Q84.The probability that a student passes in English is 0.8 and the probability that he passes in mathematics given that he has passed in English is 0.6. find the probability that he passes in both the subjects. Ans: 0.48
Q91.Among the examinees in an examination 30%, 35% and 45% failed in statistics, in Mathematics and in atleast one of the subjects respectively. An examinee is selected at random. Find the probabilities that (i) he failed I Mathematics only, (ii) he passed in statistics if it is known that he failed in Mathematics.
Q92.A company has four production sections viz. S1,S2,S3,S4 which contribute 30%, 20%, 28% and 22% respectively, to the total output. It was observed that these sections respectively produced 1%,2%,3%and 4% defective units. If a unit is selected at random and found to be defective, what is the probability that the unit so selected has come from either section one or section four
Q93.A company has two plants to manufacture scooters Plant I manufacturers 80% of the scooters and plant II manufacturers 20%.At Plant I, 85 out of 100 scooters are rated standard quality or better. At plant II, only 65 out of 100 scooters are rated standard quality or better.
i) What is the probability that the scooter selected at random came from plant I if it is known that the scooter is of standard quality?
ii) What is the probability that the scooter came from plant II if it is known that the scooter is of standard quality?
LEC 14:
Q94. X Can solve 80% of the problems while Y can solve 90% of the problems given in a statistics book. A problem is selected at random. What is the probability that at least one of them will solve the same?
Q95.Out of the numbers 1 to 120, one is selected at random. What is the probability that it is divisible by 8 or 10?
Q96.Out of five players (of which two are members of a certain club), three are to be selected to represent the country at an international tournament. Find the probability that less than two of those selected to represent are members of the club.
Q97.Three horses A,B and C are in a race. A is twice as likely to win as B, and B is twice as likely to win as C. What are the respective probabilities of winning?
Q98.There are four hotels in a certain town. If 3 men check into hotels in a day, what is the probability they check into a different hotel?
Q99.A committee of 4 people is to be appointed from 3 officers of the production department, 4 officers of the purchase department, two officers of the sales department and one chartered accountant. Find the probability of forming the committee in the following manner:
i) There must be one from each category.
ii) It should have at least one from the purchase department.
iii) The chartered accountant must be in the committee.
Q100. A box contains 4 defective and 6 goods electronic calculators. Two calculators are drawn out one by one without replacement -
i) What is the probability that the two calculators so drawn are good?
ii) One of the two calculators so drawn is tested and found to be good. What is the probability that the other one is also good?
Q101.An electronic manufacturer has two lines A and B assembling identical electronic units. The units assembled on line A are 5% defective while those assembled on line B are 10% defective. All defective units must be reworked at a significant increase in cost. During the last 8 hours shift, line A produced 200 units while line B produced 300 units. One unit is selected at random from among the 500 units produced and it is found to be defective. What is the probability that it was assembled (i) on line A? ii) on line B?
LEC 15:
Q114.If the probability that a man wins a prize of Rs.10 is 3/5 and the probability that he wins nothing is 2/5 find the mathematical expectation.
Q115.A box contains 3 red,4 green,2 black and 1 white marbles. A man is blindfolded, asked to select a marble. If he selects a red marble he gets Rs 3, for a green one, he wins Rs,2 for a black one,Rs.7 and for a white one Rs.10.What is his mathematical expectation ?
Q116.A die is tossed twice. If it shows the same number twice, Gopal gets Rs.100 otherwise he losses Rs 5.What is his mathematical expectation?
Q117.If a man purchases a raffle ticket he can win a first prize of Rs.5,000 or a second prize of Rs.2,000 with probabilities 0.001 and 0.003.What should be a fair price to pay for the ticket.?
Q118.A bag contains 2 white balls and 3 black balls. Four persons A,B,C,D in that order each draws one ball and does not replace it. The first to draw a white ball receives Rs.10.Determine their expectations.
Q119.In a business venture a man can make a profit of Rs.2,000 with a probability of 0.4 or have a loss of Rs.1000 with a probability of 0.6.What is his expected profit?
Q120.A person tosses two coins simultaneously. He receives Rs.8 for two heads, Rs.2 for one head and he is pay Rs.6 for no head. Find his expectation.
Q121.A dice is loaded in such a way that each odd number is twice as likely to occur as each even number. Find (i) the probability that the number rolled is a perfect square and (ii) the probability that the number rolled is a perfect square provided it is greater than 3.
Q122.In the play of two dice, the thrower loses if his first throw is 2,4 or 12 he wins if his first throw is a 5 or 11. Find the ratio between his probability of loosing and probability of winning in the first throw.
Q123.(a) A sample of 3 items is selected at random from a box containing 12 items of which 3 are defective. Find the possible number of defective combinations of the said 3 selected items along with probability of a defective combination.
Q124.A box contain 6 tickets. Two of the tickets carry a prize of Rs.5 each, the other four prizes are of Re.1.
If one ticket is drawn, what is the expected value of the prize?
LEC 16:
Q1.A packet of 10 electronic components is known to include 3 defectives. If 4 components are selected from the packet at random, what is the expected value of the number of defective?