PROBABILITY
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1) A bag contains 4 white 2 black balls. Two balls are drawn at random, find the probability that they are white. 2/5
2) An urn contains 9 balls, 2 of which are white, 3 blue and 4 black. 3 balls are drawn at random from the URN. What is the chance that
i) the three balls will be different colours? 2/7
ii) 2 balls will be of the same colour and the third a different colour ? 55/84
iii) three balls will be of the same colour ? 5/84
3) There are 10 persons who are to be seated around a circular table. Find the probability that two particular persons will always sit together. 2/9
4) One card is drawn from a full pack of 52 cards. Find the probability that the card drawn is
A) either a spade or a diamond. 1/2
B) either spade or a king. 1/2, 4/13
5) There are 50 tokens marked 1,2,3....50. One token is selected at random. Find the probability that it is a multiple of 4 or 6. 8/25
6) A pair of fair dice is thrown. Find the probability of getting a sum 7, When it is known that the digit in the first dice is greater than that of second. 1/5
7) The probability that a boy will not pass MBA examination is 3/5 and that a girl will not pass is 4/5 . Calculate the probability that at least one of them passes the examination. 13/25
8) An article manufactured by a company consists of two parts A and B. in the process of manufacture of part A, 9 out of 100 are likly to be defective. similarly 5 out of 100 are likely to be defective in the manufacture of part B. Calculate the probability that the assembled article will not be defective. 0.8645.
9) 8 Men in a company of 25 are graduates. If 3 men are picked out of the 25 at random, what is the probability that.
A) they are all a graduates. 14/575
B) at least one graduate. 81/115
10) A certain player, say X is known to win with probability 0.3, if the track is fast and 0.4. If the track is slow. For Monday, there is 0.7 probability of a fast track and 0.3 probability of a slow track. What is the probability that player X will win on Monday. 0.33
11) There are 4 hotels in a town. If 3 men check into hotels in a day. what is the probability that each check into a different hotel ? 3/8
12) 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 combination of the said 3 selected items along with probability of a defective combination. 0.618
13) A die is is located in such a way that each odd number is twice as likely to occur as each even number. Find
A) the probability that the number rolled is a perfect square. 1/3
B) the probability that the number rolled is a perfect square provide it is greater than 3. 1/9
14) Given below are the weekly wages (in a rupees) of six workers in a factory: 62, 90, 78, 85, 79 and 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 then the average ? 3/5
15) 100 students randomly selected from a group students are cross-classified by age and educational qualification as a below:
Qualification Age. Total
25-above 26-28 over 28
Graduate 24 19 11 54
Post Gr. 11 16 19 46
Total 35 35 30 100
A student is selected from this group. Find the probability that:
A) his age is between 26- 28 years.
0.35
B) He is a graduate. 0.54
C) his age is between 26-28 years and he is a graduate. 0.19
D) his age is between 26-28 years assuming that he is a graduate and.
0.35
E) he is a graduate assuming that his age is between 26-28 years.
0.54
16) A bag contains 4 red and 3 blue balls. Two drawings of 2 balls are made. Find the probability of drawing first 2 red balls and the second 2 blue balls.
A) If the balls are returned to the bag after the first draw; 2/49
B) if the balls are not returned after the first draw. 3/35
17) An URN contains 7 red and 4 blue balls. if two balls are drawn at random with replacement, find the probability of getting
A) 1 red and 1 blue ball. 56/121
B) two blue balls. 16/121
18) One bag contains 4 white and 2 black balls. Another bag contains 3 white and 5 black balls. If one ball is drawn from each bag. Find the probability that one is white and one is black. 13/24
19) A purse contains two silver and four gold coins. A second purse contains 4 silver and three gold coins. if a coin is taken out at random from one of the two purses, what is the probability that it is a silver coin ? 19/42
20) Two urns contains respectively 10 white, 6 red and 9 black balls and 3 white, 7 red and 15 black balls. One ball is drawn from each uen. Find the probability that:
A) both balls are red, 42/625
B) both balls are same colour.
207/625
21) A bag contains 5 red and 4 black balls. A ball is drawn at random from the bag and put into another bag which contains 3 red and 7 black balls. A ball is drawn randomly from the second bag. What is the probability that it is red ? 32/99
22) A can solve 80% of the problems given in statistics and B can solve 60% . What is the probability that at least one of them will solve a problem selected at random? 0.92
23) A problem in business statistics is given to four students A,B,C,D. their chances of solving it are 1/2, 1/3, 1/4, 1/5 respectively. What is the probability that the problem will be solved ? 4/5
24) The probability that three man hit a a target are respectively 1/6, 1/4 and 1/3. Each man shoots once at the target. What is the probability that exactly one of them hits the target ? 31/72
25) there are four letters and 4 corresponding envelopes. letters are placed one in each envelope. Find the probability that no letter goes to the corresponding addressed envelope. 3/8
26) A man has 2 French, 3 German and 4 Spanish friends. He invites one or more of them in his birthday party. Find the probability that there will be
A) at least one German friend.
448/511
B) at least one friend from each country. 315/511
27) An URN contains 2 mangoes, 3 apples and 4 oranges. Any number of fruits are selected from it. Find the probability that in the selection there will be
A) at least one mango. 40/59
B) at least one fruit of each type.
24/59.
28) 6 different books are arranged in a shelf. Find the probability that two particular books will always be together. 1/3
29) How many losses of a coin are needed so that the probability of getting at least one head is 87.5% ?
3
30) A coin is tossed 3 times in succession. Find the probability of obtaining at least two heads. 1/2
31) Two dice are rolled. Find the probability that
A) the total of the number on the die is 9 or greater;. 5/18
B) the number in the first dice is greater than that in the second die.
5/12
32) In a game of Bridge what is the probability that a hand will contains all 4 Kings. 114/4165
33) An URN contains 8 white 3 red balls. if two balls are drawn at random, find the probability that
A) both are white. 28/55
B) both are red. 3/55
C) one of each colour. 24/55
34) Find the probability of finding at least one king out of 10 cards drawn from a 52 cards. 349/595
35) From a box containing 3 blue and 5 red balls. 3 balls are drawn at random. Find the chance of getting
A) at least one red ball. 55/56
B) at most One red ball. 2/7
36) two letters are drawn from the word HOME. Find the probability that
A) both are vowels. 1/6
B) at least one is vowel. 5/6
C) One of the letter is chosen M.
1/2
37) There are 3 Geologist, 4 Engineers, 2 Statistician and 1 Doctor. A committee of 4 among them is to be formed. Find the probability that the committee
A) consists of one of each kind
B) has at least one geologist.
C) has the doctor as a member and 3 others. 4/35, 5/6, 2/5
38) A 4 digit number is formed by the digit 1, 2, 3, 4 with no repetition. Find the probability that the numbers is
A) odd. 1/2
B) divisible by 4. 1/4
39) A 5 digited number is formed by the digits 0, 1, 2, 3, 4 with no repetition. Find the probability that the number is
A) odd. 3/8
B) divisible by 4. 5/16
40) four cards are drawn from a pack of 52 cards. What is the probability that they are from four different suits? 2197/20825
41) A and B stand in a line at random with 10 other people. Find the chance that there will be 3 people between them. 4/33
42) The first 12 letters of the alphabet are written at random. Find the probability that there are exactly 4 letters between C and D.
7/66
43) The letters of the word EDUCATION are arranged at random. Find the probability that there will be exactly 4 letters between A and E. 1/9
44) The letters of the word SUNDAY are arranged at random. Find the chance that the arrangement will
A) begin with S. 1/16
B) begin with S but not end with Y
C) Vowels will occupy odd places.
2/15, 1/5
45) The letters of the word DIRECTOR are arranged at random. Find the probability that the vowels will be will be always together. 3/28
46) 6 coins are tossed. Find the probability of getting
A) exactly 2 heads. 15/64
B) at least two heads. 57/64
47) From a group of 10 players, six players are selected at random. Find the probability that Mr X and Mr Y will
A) never be in the selection. 2/15
B) always be in the selection. 1/3
48) Some fruits are selected at random from a basket containing 3 Apples, 4 mangoes and 5 oranges. Find the probability that the selection will contents.
A) at least one Apple. 90/119
B) at least one fruit of each type.
60/119
49) A man has got 2 German, 3 Spanish and 4 French friends. He invites at random some of them. Find the probability that he will invite.
A) at least one German. 384/511
B) at least one friend from each country. 45/73
50) 8 counters marked 1,,2,.....,8 four counters, are are selected at random. Find the chance of getting at least one odd and one even counter. 34/35
51) A sub-committee of 6 members is to be formed from 7 men and 4 ladies. Calculate the probability that sub committee will consist of
A) exact two ladies. 5/11
B) at least two ladies. 53/66
52) Find the chance of throwing at least 8 in a single cast with two dice. 5/12
53) In a packet of 10 watches, three are known to be defective. If two watches are selected at random from this packet, what is the probability that at least one of them is defective. 8/15
54) 40% of the student in a class are girls. If 60% and 70% of the boys and girls respectively of the class pass is certain test. What is the probability that a student selected from this class will have passed the test ? 16/25
55) In a group of 14 males and 6 females, 8 and 3 of the males and females respectively are aged about 40 years. What is the probability that a person selected at random from this group is aged above 40, given that selected person is a female? 1/2
56) In a family of four children, what is the probability that
A) all of them will have different birthdays. 364.363.362/365³
B) two of them will have the same birthday? 6.364.363/365³
57) A bag contains 5 white 4 black balls. One ball is drawn from the bag and replaced and then a second draw of a ball is made. What is the chance that the two balls drawn are of different colours? 40/81
58) A box contains 8 red and 5 white balls. Two successive draws of 3 balls are made. Find the probability that the first drawing will give 3 white balls and the second 3 red balls. if the balls are drawn
A) with replacement. 140/20449
B) without replacement. 7/429
59) Boxes 1 and 2 contain respectively 4 white, 3 red and 3 blue balls; and 5 white, 4 red and 3 blue balls. if one ball is drawn at random from each box, What is the probability that both balls are of the same colour. 41/120
60) Two boxes contain respectively 4 white and 3 red balls; and 3 white and 7 red balls. A box is chosen at random and a ball is drawn from it. Find the probability that the ball is white. 61/140
61) An URN contains 5 white and 3 black balls, and a second urn contains 4 white and 5 black balls. One of the urn is chosen at random and 2 balls are drawn from it. Find the probability that one is white and other is black. 275/504
62) Each of two identical bags contain 5 white and 5 red red balls. One ball is transferred at random from the second Bag to the first and then one ball is drawn from the first bag. Find the probability that the ball drawn is red. 1/2
63) The odds against a certain event are 5:2 and the odds in favour of another event, independent of the former are 6:5. Find the chance that at least one of the event will happen. 52/77
64) It is 8:5 against a person who is now 30 years old to live 40 years more and 3:5 in favour of a person who is now 40 years old to live 40 years more. Find the probability that at least one of these persons will be alive 40 years hence. 59/91
65) A speaks truth in 75% and B in 80% of the cases. In what percentage of cases are they likely to contradict each other in stating the same fact ? 35%
66) A person is known to hit 4 out of 5 shots. whereas another person is known to hit 3 out of 4 shots. Find the probability of hitting a target if they both try. 19/20
67) A can solve solve 80% of the problems in this book and B can solve 70%. A problem of this book is selected at random. What is the chance that the problem will be solved if they both try. 47/50
68) The probability that a student passes in Statistics test is 2/3 and the probability that he passes both Statistics and Mathematics test is 14/45. The probability that he passes at least one test is 4/5. What is the probability that he passes the Mathematics test? 4/9
69) Presuming the daily demand to be independent, you are to find the probability that over a two-day period the number of request at some service station will be
A) 9 0.48
B) 10 if the past record indicates that daily demand has either been 4 with probability 0.4 or 5 with probability 0.6. 0.36
70) Mr. X is called per an interview for 3 separate post. At the first interview there are 5 candidates, at the 2nd 4 candidates and at the 3rd 3 candidates. if the selection of each of them is equally likely, find the probability that Mr. X will be selected for the least one post. 3/5
71) A six-faced die is so biased that it is twice as likely to show an even number as an odd number when thrown. it is thrown twice. What is the probability that the sum of two numbers thrown is even. 5/9
72) In a group of equal number of men and women, 10% men and 45% women are unemployed. What is the probability that a person selected at random is employed ?
29/40
73) A problem of statistics is given to A, B and C whose chances of solving it are 1/3,1/4 and 1/5 respectively. Find the probability that the problem will be solved by at least one of them. 3/5
74) Three shots are fired at a moving target. The probability of hitting the target at the first, second and the third shot are 1/2, 3/10, and 1/10. Calculate the probability that them target gets hit. 137/200
75) The probability of A, B, C solving a problem are 1/3, 2/7 and 3/8 respectively. if all of them try to solve the problem simultaneously, find the probability that exactly one of them will solve it. 25/56
76) In a given race the odds in favour of four horses A, B, C, D are 1:3, 1:4, 1:5, 1:6 respectively. Assuming that a dead heat is impossible, find the chance that one of them wins the race. 57/140
77) A, B, and C in that order, toss a coin. The first one to throw a head wins. What are their respective chances of winning? Assuming that the game may continue indefinitely.
4/7, 2/7,1/7
78) Markmen A and B compete by taking turns to shoot at a target. Odds in favour of A hitting the target (in a single try) are 3:2 andthe odds in favour of B hitting the target (in a single try) are 4:3.
Calculate the probability of A winning the competition if he gets the first chance to shoot. 21/29
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