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Chapter 13: Probability

NCERT Solutions for CBSE Class 10 Mathematics — 100 solved questions with detailed explanations.

100

Questions

0

Topics

Solved Questions

Q1. In a Bernoulli trial with p = 0.3 and n = 5, the mean is:

1.5
1.05
0.3
3

Difficulty: Easy · Topic: Binomial Distribution

Solution

Mean of binomial distribution = np = 5 × 0.3 = 1.5.

Q2. If P(A) = 1/2, P(B) = 1/3, P(A∩B) = 1/6, then P(A|B) is:

1/2
1/3
1/6
2/3

Difficulty: Easy-Medium · Topic: Conditional Probability

Solution

P(A|B) = P(A∩B)/P(B) = (1/6)/(1/3) = 1/2.

Q3. Bayes' theorem is used to find:

Prior probability
Posterior probability
Joint probability
Marginal probability

Difficulty: Easy-Medium · Topic: Bayes' Theorem

Solution

Bayes' theorem: P(A|B) = P(B|A)P(A)/P(B). It updates prior probability to posterior probability.

Q4. E(X) for binomial distribution is:

np
npq
n/p
p/n

Difficulty: Easy-Medium · Topic: Probability

Solution

np.

Q5. Var(X) for binomial distribution is:

npq
np
np²
n²pq

Difficulty: Easy-Medium · Topic: Probability

Solution

npq.

Q6. Total probability theorem: P(A) =:

P(A)+P(B)
P(A)×P(B)
ΣP(A|B_i)P(B_i)
P(A|B)+P(A|B')

Difficulty: Easy-Medium · Topic: Probability

Solution

ΣP(A|B_i)P(B_i).

Q7. P(A)=0.1, P(B)=0.1, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.01.

Q8. P(A)=0.1, P(B)=0.1, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.19.

Q9. P(A)=0.1, P(B)=0.2, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.02.

Q10. P(A)=0.1, P(B)=0.2, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.28.

Q11. P(A)=0.1, P(B)=0.3, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.03.

Q12. P(A)=0.1, P(B)=0.3, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.37.

Q13. P(A)=0.1, P(B)=0.4, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.04.

Q14. P(A)=0.1, P(B)=0.4, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.46.

Q15. P(A)=0.1, P(B)=0.5, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.05.

Q16. P(A)=0.1, P(B)=0.5, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.55.

Q17. P(A)=0.1, P(B)=0.6, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.06.

Q18. P(A)=0.1, P(B)=0.6, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.64.

Q19. P(A)=0.1, P(B)=0.7, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.07.

Q20. P(A)=0.1, P(B)=0.7, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.73.

Q21. P(A)=0.1, P(B)=0.8, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.08.

Q22. P(A)=0.1, P(B)=0.8, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.82.

Q23. P(A)=0.1, P(B)=0.9, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.09.

Q24. P(A)=0.1, P(B)=0.9, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.91.

Q25. P(A)=0.2, P(B)=0.1, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.02.

Q26. P(A)=0.2, P(B)=0.1, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.28.

Q27. P(A)=0.2, P(B)=0.2, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.04.

Q28. P(A)=0.2, P(B)=0.2, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.36.

Q29. P(A)=0.2, P(B)=0.3, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.06.

Q30. P(A)=0.2, P(B)=0.3, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.44.

Q31. P(A)=0.2, P(B)=0.4, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.08.

Q32. P(A)=0.2, P(B)=0.4, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.52.

Q33. P(A)=0.2, P(B)=0.5, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.1.

Q34. P(A)=0.2, P(B)=0.5, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.6.

Q35. P(A)=0.2, P(B)=0.6, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.12.

Q36. P(A)=0.2, P(B)=0.6, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.68.

Q37. P(A)=0.2, P(B)=0.7, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.14.

Q38. P(A)=0.2, P(B)=0.7, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.76.

Q39. P(A)=0.2, P(B)=0.8, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.16.

Q40. P(A)=0.2, P(B)=0.8, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.84.

Q41. P(A)=0.2, P(B)=0.9, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.18.

Q42. P(A)=0.2, P(B)=0.9, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.92.

Q43. P(A)=0.3, P(B)=0.1, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.03.

Q44. P(A)=0.3, P(B)=0.1, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.37.

Q45. P(A)=0.3, P(B)=0.2, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.06.

Q46. P(A)=0.3, P(B)=0.2, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.44.

Q47. P(A)=0.3, P(B)=0.3, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.09.

Q48. P(A)=0.3, P(B)=0.3, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.51.

Q49. P(A)=0.3, P(B)=0.4, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.12.

Q50. P(A)=0.3, P(B)=0.4, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.58.

Q51. P(A)=0.3, P(B)=0.5, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.15.

Q52. P(A)=0.3, P(B)=0.5, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.65.

Q53. P(A)=0.3, P(B)=0.6, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.18.

Q54. P(A)=0.3, P(B)=0.6, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.72.

Q55. P(A)=0.3, P(B)=0.7, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.21.

Q56. P(A)=0.3, P(B)=0.7, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.79.

Q57. P(A)=0.3, P(B)=0.8, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.24.

Q58. P(A)=0.3, P(B)=0.8, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.86.

Q59. P(A)=0.3, P(B)=0.9, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.27.

Q60. P(A)=0.3, P(B)=0.9, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.93.

Q61. P(A)=0.4, P(B)=0.1, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.04.

Q62. P(A)=0.4, P(B)=0.1, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.46.

Q63. P(A)=0.4, P(B)=0.2, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.08.

Q64. P(A)=0.4, P(B)=0.2, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.52.

Q65. P(A)=0.4, P(B)=0.3, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.12.

Q66. P(A)=0.4, P(B)=0.3, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.58.

Q67. P(A)=0.4, P(B)=0.4, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.16.

Q68. P(A)=0.4, P(B)=0.4, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.64.

Q69. P(A)=0.4, P(B)=0.5, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.2.

Q70. P(A)=0.4, P(B)=0.5, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.7.

Q71. P(A)=0.4, P(B)=0.6, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.24.

Q72. P(A)=0.4, P(B)=0.6, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.76.

Q73. P(A)=0.4, P(B)=0.7, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.28.

Q74. P(A)=0.4, P(B)=0.7, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.82.

Q75. P(A)=0.4, P(B)=0.8, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.32.

Q76. P(A)=0.4, P(B)=0.8, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.88.

Q77. P(A)=0.4, P(B)=0.9, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.36.

Q78. P(A)=0.4, P(B)=0.9, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.94.

Q79. P(A)=0.5, P(B)=0.1, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.05.

Q80. P(A)=0.5, P(B)=0.1, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.55.

Q81. P(A)=0.5, P(B)=0.2, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.1.

Q82. P(A)=0.5, P(B)=0.2, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.6.

Q83. P(A)=0.5, P(B)=0.3, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.15.

Q84. P(A)=0.5, P(B)=0.3, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.65.

Q85. P(A)=0.5, P(B)=0.4, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.2.

Q86. P(A)=0.5, P(B)=0.4, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.7.

Q87. P(A)=0.5, P(B)=0.5, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.25.

Q88. P(A)=0.5, P(B)=0.5, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.75.

Q89. P(A)=0.5, P(B)=0.6, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.3.

Q90. P(A)=0.5, P(B)=0.6, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.8.

Q91. P(A)=0.5, P(B)=0.7, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.35.

Q92. P(A)=0.5, P(B)=0.7, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.85.

Q93. P(A)=0.5, P(B)=0.8, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.4.

Q94. P(A)=0.5, P(B)=0.8, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.9.

Q95. P(A)=0.5, P(B)=0.9, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.45.

Q96. P(A)=0.5, P(B)=0.9, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.95.

Q97. P(A)=0.6, P(B)=0.1, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.06.

Q98. P(A)=0.6, P(B)=0.1, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.64.

Q99. P(A)=0.6, P(B)=0.2, independent. P(A∩B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∩B)=P(A)P(B)=0.12.

Q100. P(A)=0.6, P(B)=0.2, independent. P(A∪B)=

Difficulty: Easy-Medium · Topic: Probability

Solution

P(A∪B)=P(A)+P(B)-P(A∩B)=0.68.

Other Chapters in Mathematics

Ch 1: Relations & Functions Ch 2: Inverse Trig Functions Ch 3: Matrices Ch 4: Determinants Ch 5: Continuity & Differentiability Ch 6: Applications of Derivatives Ch 7: Integrals Ch 8: Applications of Integrals Ch 9: Differential Equations Ch 10: Vector Algebra Ch 11: Three Dimensional Geometry Ch 12: Linear Programming

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