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Probability and Statistics –
High Difficulty
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#1. For data set [38, 34, 26, 22, 17, 26], what is the arithmetic mean?
#2. If X~Binomial(n=9, p=0.2), find P(X=3).
#3. For standard normal Z, estimate P(Z ≤ -1.0).
#4. Given P(A)=0.6, P(B)=0.5, and P(A∩B)=0.21, find P(A|B).
#5. A discrete random variable has P(X=1)=0.308, P(X=6)=0.077, P(X=3)=0.308, P(X=3)=0.308. Compute E[X].
#6. If T is exponential with rate λ=1.5, find P(T ≤ 3).
#7. A process has population standard deviation 9. For sample size n=64, what is the standard error of the sample mean?
#8. For X~N(μ=56, σ=9), find the z-score of x=65.
#9. Given P(A)=0.45, P(B|A)=0.7, and P(B|A^c)=0.3, compute P(A|B).
#10. If X is uniformly distributed on [7, 17], what is Var(X)?
#11. For data set [36, 18, 40, 15, 38, 27], what is the arithmetic mean?
#12. If X~Binomial(n=11, p=0.3), find P(X=4).
#13. For standard normal Z, estimate P(Z ≤ 1.5).
#14. Given P(A)=0.4, P(B)=0.2, and P(A∩B)=0.048, find P(A|B).
#15. A discrete random variable has P(X=5)=0.364, P(X=4)=0.182, P(X=6)=0.182, P(X=1)=0.273. Compute E[X].
#16. If T is exponential with rate λ=0.9, find P(T ≤ 2).
#17. A process has population standard deviation 7. For sample size n=64, what is the standard error of the sample mean?
#18. For X~N(μ=74, σ=14), find the z-score of x=60.
#19. Given P(A)=0.45, P(B|A)=0.6, and P(B|A^c)=0.2, compute P(A|B).
#20. If X is uniformly distributed on [12, 23], what is Var(X)?
#21. For data set [30, 25, 37, 13, 28, 26], what is the arithmetic mean?
#22. If X~Binomial(n=9, p=0.3), find P(X=5).
#23. Given P(A)=0.4, P(B)=0.5, and P(A∩B)=0.16, find P(A|B).
#24. A discrete random variable has P(X=6)=0.214, P(X=2)=0.286, P(X=2)=0.286, P(X=5)=0.214. Compute E[X].
#25. If T is exponential with rate λ=0.9, find P(T ≤ 4).
#26. A process has population standard deviation 4. For sample size n=33, what is the standard error of the sample mean?
#27. For X~N(μ=91, σ=11), find the z-score of x=80.
#28. Given P(A)=0.45, P(B|A)=0.7, and P(B|A^c)=0.2, compute P(A|B).
#29. If X is uniformly distributed on [12, 22], what is Var(X)?
#30. For data set [12, 25, 18, 22, 20, 38], what is the arithmetic mean?
#31. If X~Binomial(n=14, p=0.2), find P(X=4).
#32. For standard normal Z, estimate P(Z ≤ 1.0).
#33. Given P(A)=0.4, P(B)=0.2, and P(A∩B)=0.056, find P(A|B).
#34. A discrete random variable has P(X=5)=0.333, P(X=1)=0.083, P(X=6)=0.333, P(X=5)=0.25. Compute E[X].
#35. If T is exponential with rate λ=0.6, find P(T ≤ 4).
#36. A process has population standard deviation 6. For sample size n=56, what is the standard error of the sample mean?
#37. For X~N(μ=62, σ=15), find the z-score of x=77.
#38. Given P(A)=0.5, P(B|A)=0.8, and P(B|A^c)=0.2, compute P(A|B).
#39. If X is uniformly distributed on [5, 14], what is Var(X)?
#40. For data set [36, 37, 16, 13, 21, 30], what is the arithmetic mean?
#41. If X~Binomial(n=10, p=0.5), find P(X=4).
#42. For standard normal Z, estimate P(Z ≤ 0.5).
#43. Given P(A)=0.4, P(B)=0.2, and P(A∩B)=0.064, find P(A|B).
#44. A discrete random variable has P(X=2)=0.286, P(X=4)=0.214, P(X=3)=0.214, P(X=6)=0.286. Compute E[X].
#45. If T is exponential with rate λ=1.2, find P(T ≤ 1).
#46. A process has population standard deviation 5. For sample size n=39, what is the standard error of the sample mean?
#47. For X~N(μ=63, σ=8), find the z-score of x=71.
#48. Given P(A)=0.5, P(B|A)=0.7, and P(B|A^c)=0.4, compute P(A|B).
#49. If X is uniformly distributed on [7, 14], what is Var(X)?
#50. For data set [40, 37, 19, 27, 30, 10], what is the arithmetic mean?
#51. If X~Binomial(n=8, p=0.5), find P(X=4).
#52. Given P(A)=0.4, P(B)=0.4, and P(A∩B)=0.112, find P(A|B).
#53. A discrete random variable has P(X=6)=0.3, P(X=1)=0.2, P(X=3)=0.4, P(X=2)=0.1. Compute E[X].
#54. A process has population standard deviation 8. For sample size n=57, what is the standard error of the sample mean?
#55. For X~N(μ=72, σ=6), find the z-score of x=66.
#56. Given P(A)=0.45, P(B|A)=0.6, and P(B|A^c)=0.4, compute P(A|B).
#57. If X is uniformly distributed on [6, 16], what is Var(X)?
#58. For data set [31, 31, 15, 13, 17, 18], what is the arithmetic mean?
#59. If X~Binomial(n=14, p=0.4), find P(X=3).
#60. For standard normal Z, estimate P(Z ≤ -0.5).
#61. Given P(A)=0.4, P(B)=0.5, and P(A∩B)=0.12, find P(A|B).
#62. A discrete random variable has P(X=3)=0.111, P(X=2)=0.222, P(X=4)=0.444, P(X=2)=0.222. Compute E[X].
#63. If T is exponential with rate λ=0.6, find P(T ≤ 2).
#64. A process has population standard deviation 7. For sample size n=57, what is the standard error of the sample mean?
#65. For X~N(μ=56, σ=7), find the z-score of x=63.
#66. Given P(A)=0.45, P(B|A)=0.7, and P(B|A^c)=0.4, compute P(A|B).
#67. If X is uniformly distributed on [12, 20], what is Var(X)?
#68. For data set [12, 13, 39, 32, 34, 25], what is the arithmetic mean?
#69. If X~Binomial(n=12, p=0.4), find P(X=3).
#70. Given P(A)=0.6, P(B)=0.5, and P(A∩B)=0.24, find P(A|B).
#71. A discrete random variable has P(X=2)=0.3, P(X=4)=0.1, P(X=1)=0.2, P(X=4)=0.4. Compute E[X].
#72. If T is exponential with rate λ=1.2, find P(T ≤ 4).
#73. A process has population standard deviation 7. For sample size n=30, what is the standard error of the sample mean?
#74. For X~N(μ=75, σ=8), find the z-score of x=83.
#75. Given P(A)=0.6, P(B|A)=0.8, and P(B|A^c)=0.4, compute P(A|B).
#76. For data set [12, 25, 19, 36, 12, 36], what is the arithmetic mean?
#77. Given P(A)=0.6, P(B)=0.2, and P(A∩B)=0.084, find P(A|B).
#78. A discrete random variable has P(X=3)=0.125, P(X=6)=0.25, P(X=2)=0.375, P(X=3)=0.25. Compute E[X].
#79. If T is exponential with rate λ=0.9, find P(T ≤ 3).
#80. A process has population standard deviation 9. For sample size n=43, what is the standard error of the sample mean?
#81. For X~N(μ=97, σ=10), find the z-score of x=87.
#82. Given P(A)=0.6, P(B|A)=0.7, and P(B|A^c)=0.2, compute P(A|B).
#83. If X is uniformly distributed on [6, 17], what is Var(X)?
#84. For data set [36, 27, 21, 26, 35, 19], what is the arithmetic mean?
#85. If X~Binomial(n=10, p=0.2), find P(X=2).
#86. Given P(A)=0.5, P(B)=0.5, and P(A∩B)=0.175, find P(A|B).
#87. A discrete random variable has P(X=2)=0.1, P(X=2)=0.3, P(X=1)=0.3, P(X=1)=0.3. Compute E[X].
#88. If T is exponential with rate λ=1.2, find P(T ≤ 2).
#89. A process has population standard deviation 6. For sample size n=39, what is the standard error of the sample mean?
#90. For X~N(μ=68, σ=10), find the z-score of x=58.
#91. Given P(A)=0.45, P(B|A)=0.8, and P(B|A^c)=0.4, compute P(A|B).
#92. If X is uniformly distributed on [9, 16], what is Var(X)?
#93. For data set [29, 26, 21, 40, 26, 17], what is the arithmetic mean?
#94. If X~Binomial(n=12, p=0.2), find P(X=2).
#95. Given P(A)=0.6, P(B)=0.3, and P(A∩B)=0.144, find P(A|B).
#96. A discrete random variable has P(X=2)=0.444, P(X=3)=0.222, P(X=6)=0.111, P(X=3)=0.222. Compute E[X].
#97. If T is exponential with rate λ=0.9, find P(T ≤ 1).
#98. A process has population standard deviation 7. For sample size n=45, what is the standard error of the sample mean?
#99. For X~N(μ=85, σ=5), find the z-score of x=80.
#100. Given P(A)=0.55, P(B|A)=0.7, and P(B|A^c)=0.2, compute P(A|B).
#102. For data set [38, 34, 26, 22, 17, 26], what is the arithmetic mean?
#103. If X~Binomial(n=9, p=0.2), find P(X=3).
#104. For standard normal Z, estimate P(Z ≤ -1.0).
#105. Given P(A)=0.6, P(B)=0.5, and P(A∩B)=0.21, find P(A|B).
#106. A discrete random variable has P(X=1)=0.308, P(X=6)=0.077, P(X=3)=0.308, P(X=3)=0.308. Compute E[X].
#107. If T is exponential with rate λ=1.5, find P(T ≤ 3).
#108. A process has population standard deviation 9. For sample size n=64, what is the standard error of the sample mean?
#109. For X~N(μ=56, σ=9), find the z-score of x=65.
#110. Given P(A)=0.45, P(B|A)=0.7, and P(B|A^c)=0.3, compute P(A|B).
#111. If X is uniformly distributed on [7, 17], what is Var(X)?
#112. For data set [36, 18, 40, 15, 38, 27], what is the arithmetic mean?
#113. If X~Binomial(n=11, p=0.3), find P(X=4).
#114. For standard normal Z, estimate P(Z ≤ 1.5).
#115. Given P(A)=0.4, P(B)=0.2, and P(A∩B)=0.048, find P(A|B).
#116. A discrete random variable has P(X=5)=0.364, P(X=4)=0.182, P(X=6)=0.182, P(X=1)=0.273. Compute E[X].
#117. If T is exponential with rate λ=0.9, find P(T ≤ 2).
#118. A process has population standard deviation 7. For sample size n=64, what is the standard error of the sample mean?
#119. For X~N(μ=74, σ=14), find the z-score of x=60.
#120. Given P(A)=0.45, P(B|A)=0.6, and P(B|A^c)=0.2, compute P(A|B).
#121. If X is uniformly distributed on [12, 23], what is Var(X)?
#122. For data set [30, 25, 37, 13, 28, 26], what is the arithmetic mean?
#123. If X~Binomial(n=9, p=0.3), find P(X=5).
#124. Given P(A)=0.4, P(B)=0.5, and P(A∩B)=0.16, find P(A|B).
#125. A discrete random variable has P(X=6)=0.214, P(X=2)=0.286, P(X=2)=0.286, P(X=5)=0.214. Compute E[X].
#126. If T is exponential with rate λ=0.9, find P(T ≤ 4).
#127. A process has population standard deviation 4. For sample size n=33, what is the standard error of the sample mean?
#128. For X~N(μ=91, σ=11), find the z-score of x=80.
#129. Given P(A)=0.45, P(B|A)=0.7, and P(B|A^c)=0.2, compute P(A|B).
#130. If X is uniformly distributed on [12, 22], what is Var(X)?
#131. For data set [12, 25, 18, 22, 20, 38], what is the arithmetic mean?
#132. If X~Binomial(n=14, p=0.2), find P(X=4).
#133. For standard normal Z, estimate P(Z ≤ 1.0).
#134. Given P(A)=0.4, P(B)=0.2, and P(A∩B)=0.056, find P(A|B).
#135. A discrete random variable has P(X=5)=0.333, P(X=1)=0.083, P(X=6)=0.333, P(X=5)=0.25. Compute E[X].
#136. If T is exponential with rate λ=0.6, find P(T ≤ 4).
#137. A process has population standard deviation 6. For sample size n=56, what is the standard error of the sample mean?
#138. For X~N(μ=62, σ=15), find the z-score of x=77.
#139. Given P(A)=0.5, P(B|A)=0.8, and P(B|A^c)=0.2, compute P(A|B).
#140. If X is uniformly distributed on [5, 14], what is Var(X)?
#141. For data set [36, 37, 16, 13, 21, 30], what is the arithmetic mean?
#142. If X~Binomial(n=10, p=0.5), find P(X=4).
#143. For standard normal Z, estimate P(Z ≤ 0.5).
#144. Given P(A)=0.4, P(B)=0.2, and P(A∩B)=0.064, find P(A|B).
#145. A discrete random variable has P(X=2)=0.286, P(X=4)=0.214, P(X=3)=0.214, P(X=6)=0.286. Compute E[X].
#146. If T is exponential with rate λ=1.2, find P(T ≤ 1).
#147. A process has population standard deviation 5. For sample size n=39, what is the standard error of the sample mean?
#148. For X~N(μ=63, σ=8), find the z-score of x=71.
#149. Given P(A)=0.5, P(B|A)=0.7, and P(B|A^c)=0.4, compute P(A|B).
#150. If X is uniformly distributed on [7, 14], what is Var(X)?
#151. For data set [40, 37, 19, 27, 30, 10], what is the arithmetic mean?
#152. If X~Binomial(n=8, p=0.5), find P(X=4).
#153. Given P(A)=0.4, P(B)=0.4, and P(A∩B)=0.112, find P(A|B).
#154. A discrete random variable has P(X=6)=0.3, P(X=1)=0.2, P(X=3)=0.4, P(X=2)=0.1. Compute E[X].
#155. A process has population standard deviation 8. For sample size n=57, what is the standard error of the sample mean?
#156. For X~N(μ=72, σ=6), find the z-score of x=66.
#157. Given P(A)=0.45, P(B|A)=0.6, and P(B|A^c)=0.4, compute P(A|B).
#158. If X is uniformly distributed on [6, 16], what is Var(X)?
#159. For data set [31, 31, 15, 13, 17, 18], what is the arithmetic mean?
#160. If X~Binomial(n=14, p=0.4), find P(X=3).
#161. For standard normal Z, estimate P(Z ≤ -0.5).
#162. Given P(A)=0.4, P(B)=0.5, and P(A∩B)=0.12, find P(A|B).
#163. A discrete random variable has P(X=3)=0.111, P(X=2)=0.222, P(X=4)=0.444, P(X=2)=0.222. Compute E[X].
#164. If T is exponential with rate λ=0.6, find P(T ≤ 2).
#165. A process has population standard deviation 7. For sample size n=57, what is the standard error of the sample mean?
#166. For X~N(μ=56, σ=7), find the z-score of x=63.
#167. Given P(A)=0.45, P(B|A)=0.7, and P(B|A^c)=0.4, compute P(A|B).
#168. If X is uniformly distributed on [12, 20], what is Var(X)?
#169. For data set [12, 13, 39, 32, 34, 25], what is the arithmetic mean?
#170. If X~Binomial(n=12, p=0.4), find P(X=3).
#171. Given P(A)=0.6, P(B)=0.5, and P(A∩B)=0.24, find P(A|B).
#172. A discrete random variable has P(X=2)=0.3, P(X=4)=0.1, P(X=1)=0.2, P(X=4)=0.4. Compute E[X].
#173. If T is exponential with rate λ=1.2, find P(T ≤ 4).
#174. A process has population standard deviation 7. For sample size n=30, what is the standard error of the sample mean?
#175. For X~N(μ=75, σ=8), find the z-score of x=83.
#176. Given P(A)=0.6, P(B|A)=0.8, and P(B|A^c)=0.4, compute P(A|B).
#177. For data set [12, 25, 19, 36, 12, 36], what is the arithmetic mean?
#178. Given P(A)=0.6, P(B)=0.2, and P(A∩B)=0.084, find P(A|B).
#179. A discrete random variable has P(X=3)=0.125, P(X=6)=0.25, P(X=2)=0.375, P(X=3)=0.25. Compute E[X].
#180. If T is exponential with rate λ=0.9, find P(T ≤ 3).
#181. A process has population standard deviation 9. For sample size n=43, what is the standard error of the sample mean?
#182. For X~N(μ=97, σ=10), find the z-score of x=87.
#183. Given P(A)=0.6, P(B|A)=0.7, and P(B|A^c)=0.2, compute P(A|B).
#184. If X is uniformly distributed on [6, 17], what is Var(X)?
#185. For data set [36, 27, 21, 26, 35, 19], what is the arithmetic mean?
#186. If X~Binomial(n=10, p=0.2), find P(X=2).
#187. Given P(A)=0.5, P(B)=0.5, and P(A∩B)=0.175, find P(A|B).
#188. A discrete random variable has P(X=2)=0.1, P(X=2)=0.3, P(X=1)=0.3, P(X=1)=0.3. Compute E[X].
#189. If T is exponential with rate λ=1.2, find P(T ≤ 2).
#190. A process has population standard deviation 6. For sample size n=39, what is the standard error of the sample mean?
#191. For X~N(μ=68, σ=10), find the z-score of x=58.
#192. Given P(A)=0.45, P(B|A)=0.8, and P(B|A^c)=0.4, compute P(A|B).
#193. If X is uniformly distributed on [9, 16], what is Var(X)?
#194. For data set [29, 26, 21, 40, 26, 17], what is the arithmetic mean?
#195. If X~Binomial(n=12, p=0.2), find P(X=2).
#196. Given P(A)=0.6, P(B)=0.3, and P(A∩B)=0.144, find P(A|B).
#197. A discrete random variable has P(X=2)=0.444, P(X=3)=0.222, P(X=6)=0.111, P(X=3)=0.222. Compute E[X].
#198. If T is exponential with rate λ=0.9, find P(T ≤ 1).
#199. A process has population standard deviation 7. For sample size n=45, what is the standard error of the sample mean?
#200. For X~N(μ=85, σ=5), find the z-score of x=80.
#201. Given P(A)=0.55, P(B|A)=0.7, and P(B|A^c)=0.2, compute P(A|B).
