Probability

 

1. The outcome of tossing a coin is simple event.
2. Classical probability is measured in terms of a ratio.
3. Probability can take values 0 to 1. It is expressed as percentage, proportion.
4. If one does not affect the occurrence of the other, then it said to be two events.
5. If A and B are two events which have no point in common, the events A and B are mutually exclusive.
6. Classical probability is also known as Laplace's probability.
7. Each outcome of a random experiment is called Primary event.
8. If A and B are two events, the probability of occurrence of either A or B is given as P (A U B).
9. The limiting relative frequency approach of probability is known as statistical probability.
10. The definition of statistical probability was originally given by Laplace.
11. If it is known that an event A has occurred, the probability of an event E given A is called conditional probability.
12. Probability by classical approach has many lecunae.
13. Classical probability is possible in case of equilikely outcomes.
14. An event consisting of those elements which are not in A is called as complementary event.
15. The probability of all possible outcomes of a random experiment is always equal to one.
16. The probability of the intersection of two mutually exclusive events is always zero.
17. The individual probabilities of occurrence of two events A and B are known, the probability of occurrence of both the events together will be decreased.
18. If two events A and B are such that A subset B and B subset A, the relation between P(A) and P(B) is P(A) = P(B).
19. If A is an event, the conditional probability of A given A is equal to one.
20. If A subset B, the probability, P (A/B) is equals to P(A) / P(B).
21. If B subset A, the probability, P (A/B) is equal to one.
22. If two events A and B are such that A subset B, the relation between the conditional probability P (A/C) and P(B/C) is P(A/C) < P(B/C).
23. For any two events A and B, P (A - B) is equal to P (A) - P (AB).
24. If an event B has occurred and it is known that P (B) = 1, the conditional probability P (A/B) is equal to P (A).
25. If A and B are any two mutually exclusive events, the P (A/A U B) is equal to P(A) / [P (A) + P (B)].
26. The idea of posteriori probabilities was introduced by Thomas Bayes.
27. In a city 60 percent read newspaper A, 40 % read newspaper B and 30 % read newspaper C, 20 % read A and B, 30 % read A and C, 10 % read B and C. Also 15 % read papers A, B, C. The percentage who does not read any these newspapers is 15%.
28. If a bag contains 4 white and 3 black balls. Two draws of 2 balls are successively made, the probability of getting 2 white balls at first draw and 2 black balls at second draw when the balls drawn at first draw were replaced is 2/49. If the balls are not replaced after the first draw, the probability of 2 black balls at second draw is 3/35.
29. In tossing 3 coins at a time, the probability of getting at most one head is 1/2.
30. There is 80 % chance that a problem will be solved by a statistics student and 60 % chance is there that the same problem will be solved by the mathematics student. The probability that at least the problem will be solved is 0.92.
31. The probability of two persons being borned on the same day is 1/7.
32. An urn contains 5 red, 4 white and 3 black balls. The probability of 3 balls being of different colors when the ball is replaced after each draw is equal to 2/144. The probability of 3 balls being drawn in the order red, white and black when the balls are not replaced after each draw, is equal to 1/22.
33. An urn A contains 5 white, and 3 black balls and B contains 4 white and 4 black balls. An urn is selected, and a ball is drawn from it, the probability, that the ball is white, is 5/16.
34. From a pack of 52 cards, two cards are drawn at random. The probability that one is an ace and the other is a king is 8/663.
35. Two dice are rolled by two players A and B. A throw 10, the probability that B throws more than A is 1/12.
36. The data reveals that 10 % patients die in a particular type of operation. A doctor performed 9 operations and all of them survived. Whether the 10th patient on being operated may survive or die.
37. There are two groups of students consisting of 4 boys and 2 girls: 3 boys and 1 girl. One student is selected from both the groups. The probability of one boy and one girl being selected is 5/12.
38. In a shooting competition, Mr. X can shoot at the bull's eye 4 times out of 5 shots and Mr. Y, 5 times out of six and Mr. Z, 3 times out of 4 shots. The probability that the target will be hit at least twice is 107/120.
39. There are two bags. One bag contains 4 red and 5 black balls and the other 5 red and 4 black balls. One ball is to be drawn from either of the two bags. The probability of drawing a black ball is 1/2.
40. 3 dice are rolled simultaneously. The probability of getting 12 spots is 25/216.
41. A bag contains 3 white and 5 red balls. 3 balls are drawn after shaking the bag. The odds against these balls being red is 5/28.
42. A bag contains 3 white, 1 black and 3 red balls. Two balls are drawn from the well shaked bag. The probability of both the balls being black is zero.
43. The chance of winning the race of the horse A in Durby is 1/5 and that of horse B is 1/. The probability that the race will be won by A or B is 11/30.
44. Four cards are drawn from a pack of 52 cards. The probability that out of 4 cards being 2 red and 2 black is 325/833.
45. For a 60-year-old person living up to the age of 70, it is 7:5 against him and for another 70-year-old person surviving up to the age of 80, it is 5:2 against him. The probability that one of them will survive for 10 years more is 49/84.
46. If 7:6 is in favour of A to survive 5 years more and 5:3 in favour of B to survive 5 years more, the probability that at least one of them will survive for 5 years more is 43/52.
47. The probability of throwing an odd sum with two fair dice is 1/2.
48. The probability that there is at least one spot in two rollings of a die is 11/36.
49. A group consists of 4 men, 3 women and 2 boys. 3 persons are selected at random. The probability that 2 men are selected is 5/14.
50. The probability that a leap year will have 53 sundays is 2/7.
51. If the chance of a A hiting a target is 3 times out of 4 and of B 4 times out of 5 and of C 5 times out of 6. The probability that the target will be hit in two hits is 47/120.
52. A consignment of 15 record players contains 4 defectives. The record players are selected at random one by one and examined. The ones examined are not put back. The probability that the 9th piece examined is the last defective one is 8/195.
53. The chance that doctors A will diagnose a disease X correctly is 60 %. The chance that a patient will die by this treatment after correct diagnosis is 40 %, and the chance of death by wrong diagnosis is 70%. A patient of doctor A, who had disease X, died. The probability that his disease was diagnosed correctly is 6/13.
54. An urn contains 5 yellow, 4 black and 3 white balls. 3 balls are drawn at random. The probability that no black ball is selected is 7/55.
55. A bag contains 3 white and 5 red balls. A game is played such that a ball is drawn, its color is noted and replaced with two additional balls of the same color. The selection is made 3 times. The probability that a white ball is selected at each trial is 7/64.
56. A can hit a target 2 times in 5 shots, B 3 times in 5 shots and C 4 times in 5 shots. They fire a volley. The probability that two shots hit is 58/125.
57. There are four coins in a bag. One of the coins has head on both sides. A coin is drawn at random and tossed five times and fell always with head upward. The probability that it is the coin with two head is 32/35.
58. One of the two events is certain to happen. The chance of one event is one-fifth of the other. The odds in favour of the other is 1:5.
59. One of the two events must happen; given that the chance of one is one-fourth of the other. The odd is favour of the other is 1:4.
60. A coin is tossed six times. The probability of obtaining heads and tails alternately is 1/32.
61. The odds in favour of the certain event are 5:4, and odds against another event are 4:3. The chance that at least one of them will happen is 47/63.
62. A and B start in a ring with ten other persons. If the arrangement of 12 persons is a random, the chance that there are exactly 3 persons between A and B is 2/11.
63. 3 houses were available in a locality for allotment. 3 persons applied for a house. The probablity that all the 3 persons applied for the same house is 1/9.
64. A speaks truth 4 times out of five and B speaks truth 3 times out of 4. They agree in the assertion that a white ball has been drawn from a bag containing 10 balls of different colours. The probability that a white ball was really drawn is 81/82.
65. The probability of drawing a white ball in the first draw and again a white ball in the second draw with replacement from a bag containing 6 white and 4 blue balls is 36/100.
66. A fair coin is tossed repeatedly unless a head is obtained. The probability that the coin has to be tossed at least 4 times is 1/4.
67. Out of 20 employees in a company, 5 are graduates. 3 employees are selected at random. The probability of all the 3 being graduates is 1/114.
68. A machine's part is produced by 3 factories A, B and C. Their proportional production is 25, 35 and 40 %, respectively. Also, the percentage defectives manufactured by 3 factories are 5,4 and 3, respectively. A part is taken at random and is found to be defective. The probability that the selected part belongs to factor B is 4/11.
69. A card is drawn from a well shuffled pack of 52 cards. A gambler bets that it is either a heart or an ace. What are odds against his winning this wet as 9:4.
70. An unbiased coin is tossed 4 times. The probability that the number of heads exceeds the number of tails is 5/16.