集合符号练习题 - 掌握集合运算和概率计算
以下是5道综合练习题，涵盖集合符号的理解、维恩图分析和概率计算等核心内容。
Use set notation to describe the shaded region in each Venn diagram:
a) Shaded only \( A \)
b) Shaded \( B \cup A' \)
c) Shaded \( (A \cap B)' \)
d) Shaded \( A \cap B \cap C' \)
解答过程：
a) 仅A区域：\( A \cap B' \)
b) B或非A：\( B \cup A' \)
c) 非(A且B)：\( (A \cap B)' = A' \cup B' \)
d) A且B且非C：\( A \cap B \cap C' \)
A bag contains 30 marbles: 12 red only, 5 blue only, 3 red and blue, and 10 green. Let \( R \) be "red" and \( B \) be "blue". Find:
a) \( P(R \cap B) \)
b) \( P(R \cup B) \)
c) \( P(R' \cap G) \)（\( G \)为"green"）
解答过程：
a) 红且蓝的概率
\( P(R \cap B) = \frac{3}{30} = 0.1 \)
b) 红或蓝的概率
\( P(R \cup B) = \frac{12 + 5 + 3}{30} = \frac{20}{30} = \frac{2}{3} \)
c) 非红且绿的概率
绿色的总数：10张，全部是非红的，故 \( P(R' \cap G) = \frac{10}{30} = \frac{1}{3} \)
Events \( X \) and \( Y \) are such that \( P(X) = 0.6 \), \( P(Y) = 0.3 \), and \( P(X \cap Y) = 0.18 \). Determine if \( X \) and \( Y \) are independent.
解答过程：
若\( X \)、\( Y \)独立，需满足 \( P(X \cap Y) = P(X) \times P(Y) = 0.6 \times 0.3 = 0.18 \)，与题目一致，故**\( X \)和\( Y \)独立**。
Events \( A \) and \( B \) are mutually exclusive, \( P(A) = 0.2 \), \( P(B) = 0.5 \). Find:
a) \( P(A \cup B) \)
b) \( P(A' \cap B') \)
解答过程：
a) 互斥事件并集概率
\( P(A \cup B) = 0.2 + 0.5 = 0.7 \)
b) 既非A也非B的概率
\( P(A' \cap B') = 1 - P(A \cup B) = 1 - 0.7 = 0.3 \)
A class of 40 students: 25 study Maths (\( M \)), 20 study Physics (\( P \)), 15 study both. Find:
a) \( P(M \cap P) \)
b) \( P(M \cup P) \)
c) \( P(M' \cap P') \)
解答过程：
a) 数学且物理的概率
\( P(M \cap P) = \frac{15}{40} = 0.375 \)
b) 数学或物理的概率
\( P(M \cup P) = \frac{25 + 20 - 15}{40} = \frac{30}{40} = 0.75 \)
c) 既不数学也不物理的概率
\( P(M' \cap P') = \frac{40 - 30}{40} = \frac{10}{40} = 0.25 \)