We Learn Maths

We Learn Maths

NCERT (Class 12)

IDProblem
1Find the Chance
2Find the Probability of Receiving a Package
3Drawing a Card
  2. In a deck of 52 cards, what is the probability of drawing either a King or a Queen?
  3. A die is rolled. What is the probability of getting either a 2 or a 5?
  4. In a class of 30 students, 10 are in the drama club and 15 are in the sports club. If a student is selected at random, what is the probability that the student is either in the drama club or the sports club, assuming no student is in both clubs?
  5. A box contains 5 red balls and 7 blue balls. What is the probability of picking either a red ball or a blue ball?
  6. In a survey, 40% of respondents prefer tea and 60% prefer coffee. What is the probability that a randomly selected respondent prefers either tea or coffee, assuming no one prefers both?
  7. A coin is flipped. What is the probability of getting either heads or tails?
  8. In a lottery, a ticket can either win a prize or not win a prize. What is the probability of a ticket winning or not winning?
  9. In a multiple-choice question with four options (A, B, C, D), what is the probability of selecting either option A or option B?
  10. A student can either pass or fail an exam. What is the probability of a student passing or failing the exam?
  11. In a raffle, a participant can either win the first prize or the second prize. What is the probability of winning either the first prize or the second prize, assuming one cannot win both?
  12. In a deck of 52 cards, what is the probability of not drawing an Ace?
  13. A die is rolled. What is the probability of not getting a 6?
  14. In a class of 30 students, 18 are girls. If a student is selected at random, what is the probability that the student is not a girl?
  15. A box contains 8 red balls and 12 blue balls. What is the probability of not picking a red ball?
  16. In a survey, 70% of respondents prefer chocolate ice cream. What is the probability that a randomly selected respondent does not prefer chocolate ice cream?
  17. A coin is flipped. What is the probability of not getting heads?
  18. In a lottery, the probability of winning a prize is 0.2. What is the probability of not winning a prize?
  19. In a multiple-choice question with four options (A, B, C, D), what is the probability of not selecting option A?
  20. A student has a 75% chance of passing an exam. What is the probability that the student does not pass the exam?
  21. In a raffle, the probability of winning the first prize is 0.05. What is the probability of not winning the first prize?
  22. A bag contains 5 red balls and 3 blue balls. Two balls are drawn one after the other without replacement. What is the probability that both balls drawn are red?
  23. A box contains 6 green and 4 yellow marbles. Two marbles are drawn one after the other with replacement. What is the probability that the first marble is green and the second marble is yellow?
  24. A card is drawn from a deck of 52 cards, and then a second card is drawn without replacing the first. What is the probability that both cards drawn are aces?
  25. In a class of 20 students, 12 are boys and 8 are girls. Two students are selected one after the other without replacement. What is the probability that the first student selected is a boy and the second is a girl?
  26. A die is rolled twice. What is the probability of getting a 4 on the first roll and an even number on the second roll?
  27. A box contains 10 pens, 4 of which are defective. Two pens are selected one after the other without replacement. What is the probability that both pens selected are defective?
  28. A coin is flipped twice. What is the probability of getting heads on the first flip and tails on the second flip?
  29. In a survey, 60% of respondents prefer tea and 40% prefer coffee. If two respondents are selected one after the other with replacement, what is the probability that both prefer tea?
  30.  A jar contains 5 red and 7 blue candies. Two candies are drawn one after the other without replacement. What is the probability that the first candy is blue and the second candy is red?
  31. A student has a 0.7 probability of passing the first exam and a 0.8 probability of passing the second exam, assuming the outcomes are independent. What is the probability that the student passes both exams?
  32. A factory produces three types of widgets: A, B, and C. The probabilities of producing each type are 0.3, 0.5, and 0.2, respectively. The probability of a defective widget is 0.01 for type A, 0.02 for type B, and 0.03 for type C. What is the probability that a randomly selected widget is defective?
  33. An insurance company classifies drivers into three categories: low-risk, medium-risk, and high-risk, with probabilities of 0.6, 0.3, and 0.1, respectively. The probabilities of filing a claim in a year are 0.02 for low-risk, 0.05 for medium-risk, and 0.1 for high-risk drivers. What is the probability that a randomly selected driver files a claim in a year?
  34. A student can choose one of three subjects: Math, Science, or English, with probabilities of 0.4, 0.35, and 0.25, respectively. The probabilities of passing the exam are 0.9 for Math, 0.8 for Science, and 0.7 for English. What is the probability that the student passes the exam?
  35. A company has three machines: X, Y, and Z, which produce 40%, 35%, and 25% of the total output, respectively. The probabilities of a defective item are 0.02 for machine X, 0.03 for machine Y, and 0.04 for machine Z. What is the probability that a randomly selected item is defective?
  36. A hospital has three departments: Cardiology, Neurology, and Orthopedics, with probabilities of admitting patients being 0.5, 0.3, and 0.2, respectively. The probabilities of a patient recovering are 0.85 for Cardiology, 0.75 for Neurology, and 0.65 for Orthopedics. What is the probability that a randomly admitted patient recovers?
  37. A marketing survey categorizes customers into three groups: Regular, Occasional, and Rare, with probabilities of 0.5, 0.3, and 0.2, respectively. The probabilities of making a purchase are 0.6 for Regular, 0.4 for Occasional, and 0.2 for Rare customers. What is the probability that a randomly selected customer makes a purchase?
  38. A software company has three types of projects: Small, Medium, and Large, with probabilities of 0.4, 0.4, and 0.2, respectively. The probabilities of completing the project on time are 0.9 for Small, 0.8 for Medium, and 0.7 for Large projects. What is the probability that a randomly selected project is completed on time?
  39. A university offers three types of scholarships: Merit-based, Need-based, and Athletic, with probabilities of 0.5, 0.3, and 0.2, respectively. The probabilities of a student accepting the scholarship are 0.95 for Merit-based, 0.85 for Need-based, and 0.75 for Athletic scholarships. What is the probability that a randomly selected student accepts the scholarship?
  40. A retailer sells three brands of laptops: Brand A, Brand B, and Brand C, with probabilities of 0.4, 0.35, and 0.25, respectively. The probabilities of a laptop being returned are 0.05 for Brand A, 0.07 for Brand B, and 0.1 for Brand C. What is the probability that a randomly sold laptop is returned?
  41. A bank offers three types of loans: Personal, Home, and Auto, with probabilities of 0.5, 0.3, and 0.2, respectively. The probabilities of a loan being approved are 0.9 for Personal, 0.85 for Home, and 0.8 for Auto loans. What is the probability that a randomly selected loan is approved?
  42. A factory produces 60% of its products from Machine A and 40% from Machine B. The defect rates are 3% for Machine A and 5% for Machine B. If a product is found to be defective, what is the probability that it was produced by Machine A?
  43. In a medical test, the probability of having a disease is 0.01. The test has a 99% sensitivity (true positive rate) and a 98% specificity (true negative rate). If a person tests positive, what is the probability that they actually have the disease?
  44. A company has two suppliers, X and Y. Supplier X provides 70% of the parts with a defect rate of 2%, while Supplier Y provides 30% of the parts with a defect rate of 4%. If a part is found to be defective, what is the probability that it came from Supplier X?
  45. In a class, 40% of students are boys and 60% are girls. The probability of a boy passing an exam is 0.8, and the probability of a girl passing the exam is 0.9. If a student is known to have passed the exam, what is the probability that the student is a girl?
  46. A bag contains 5 red and 7 blue balls. A ball is drawn at random, and its color is noted. The ball is then replaced, and another ball is drawn. If the second ball is red, what is the probability that the first ball was also red?
  47. A diagnostic test for a certain disease has a 95% accuracy rate. The prevalence of the disease in the population is 0.5%. If a person tests positive, what is the probability that they actually have the disease?
  48. A company has three machines: A, B, and C, which produce 50%, 30%, and 20% of the total output, respectively. The defect rates are 1% for Machine A, 2% for Machine B, and 3% for Machine C. If a product is found to be defective, what is the probability that it was produced by Machine B?
  49. In a survey, 30% of respondents are from urban areas, and 70% are from rural areas. The probability of an urban respondent supporting a policy is 0.6, and the probability of a rural respondent supporting the policy is 0.4. If a respondent is known to support the policy, what is the probability that they are from an urban area?
  50. A student has a 20% chance of being late to class. If the student is late, there is a 70% chance that they missed the bus. If the student is not late, there is a 10% chance that they missed the bus. If the student missed the bus, what is the probability that they were late to class?
  51. A factory has two production lines, A and B. Line A produces 40% of the items with a defect rate of 1%, while Line B produces 60% of the items with a defect rate of 2%. If an item is found to be defective, what is the probability that it was produced by Line A?
  52.  A box contains 3 red, 4 blue, and 5 green balls. Two balls are drawn at random without replacement. What is the probability that the first ball is red given that the second ball is blue?
  53. In a class of 10 boys and 15 girls, 5 boys and 8 girls are selected for a project. If a student is chosen at random, what is the probability that the student is a boy given that they are part of the project team?
  54. A deck of 52 cards is shuffled, and two cards are drawn one after the other without replacement. What is the probability that the first card is an Ace given that the second card is a King?
  55. A committee of 5 members is to be formed from a group of 7 men and 6 women. If a committee is selected at random, what is the probability that it includes exactly 3 men given that it includes at least 2 women?
  56. In a lottery, there are 10 winning tickets and 90 losing tickets. If two tickets are drawn at random without replacement, what is the probability that the first ticket is a winner given that the second ticket is a loser?
  57. A bag contains 6 red, 4 blue, and 5 yellow marbles. Three marbles are drawn at random without replacement. What is the probability that the first marble is red given that the third marble is yellow?
  58. In a group of 8 men and 7 women, a team of 4 members is to be selected. If a team is chosen at random, what is the probability that it includes at least 2 men given that it includes exactly 1 woman?
  59. A box contains 5 defective and 15 non-defective items. Two items are drawn at random without replacement. What is the probability that the first item is defective given that the second item is non-defective?
  60. A deck of 52 cards is shuffled, and three cards are drawn one after the other without replacement. What is the probability that the first card is a Queen given that the third card is a Jack?
  61. In a class of 12 students, 7 are boys and 5 are girls. If two students are selected at random without replacement, what is the probability that the first student is a girl given that the second student is a boy?
  62. A bag contains 4 red balls and 6 blue balls. Two balls are drawn one after the other without replacement. What is the probability that the second ball is blue given that the first ball is red?
  63. A deck of 52 cards is shuffled, and two cards are drawn one after the other without replacement. What is the probability that the second card is a King given that the first card is a Queen?
  64. In a class of 20 students, 12 are boys and 8 are girls. Two students are selected one after the other without replacement. What is the probability that the second student is a girl given that the first student is a boy?
  65. A box contains 5 defective and 15 non-defective items. Two items are drawn one after the other without replacement. What is the probability that the second item is non-defective given that the first item is defective?
  66. A die is rolled twice. What is the probability of getting a 5 on the second roll given that the first roll was a 3?
  67. A jar contains 6 red, 4 blue, and 5 yellow marbles. Two marbles are drawn one after the other without replacement. What is the probability that the second marble is yellow given that the first marble is blue?
  68. In a group of 10 men and 15 women, two people are selected one after the other without replacement. What is the probability that the second person is a woman given that the first person is a man?
  69. A box contains 8 red and 12 blue balls. Two balls are drawn one after the other without replacement. What is the probability that the second ball is red given that the first ball is blue?
  70. A deck of 52 cards is shuffled, and three cards are drawn one after the other without replacement. What is the probability that the third card is a Jack given that the first card is a 10 and the second card is a 9?
  71. In a class of 30 students, 18 are girls and 12 are boys. Two students are selected one after the other without replacement. What is the probability that the second student is a boy given that the first student is a girl?
  78. Scenario: You own a candy factory that produces two types of candies: ChocoDelight and FruitChew. Each type of candy requires a certain amount of sugar and cocoa. You have a limited supply of both sugar and cocoa, and you want to maximize your profit.
    Details:
    Each ChocoDelight requires 2 units of sugar and 3 units of cocoa.
    Each FruitChew requires 4 units of sugar and 2 units of cocoa.
    You have a total of 100 units of sugar and 90 units of cocoa available.
    The profit from each ChocoDelight is $5.
    The profit from each FruitChew is $4.
    Objective: 
    Determine the number of ChocoDelight and FruitChew candies to produce in order to maximize your profit, while not exceeding the available resources.
    Hint:
  79. Scenario: A farmer has a piece of land that he wants to use to plant two types of crops: Wheat and Corn. Each type of crop requires a certain amount of water and fertilizer. The farmer has a limited supply of both water and fertilizer and wants to maximize his profit.
    Details:
    Each acre of Wheat requires 3 units of water and 2 units of fertilizer.
    Each acre of Corn requires 4 units of water and 3 units of fertilizer.
    The farmer has a total of 120 units of water and 90 units of fertilizer available.
    The profit from each acre of Wheat is $50.
    The profit from each acre of Corn is $40.
    Objective: Determine the number of acres of Wheat and Corn to plant in order to maximize the farmer’s profit, while not exceeding the available resources.