维恩图中条件概率练习题 - 掌握图形化条件概率计算
以下是3道精选练习题，涵盖维恩图中条件概率的各种计算技巧和复杂情境。
The Venn diagram shows the probabilities for two events, \( A \) and \( B \).
Find:
a) \( P(A \cup B) \)
b) \( P(A|B) \)
c) \( P(B|A') \)
d) \( P(B|A \cup B) \)
(Venn diagram: Only \( A = 0.3 \), \( A \cap B = 0.12 \), Only \( B = 0.28 \), Neither = 0.3)
解答过程：
a) \( P(A \cup B) = 0.3 + 0.12 + 0.28 = 0.7 \)
b) \( P(A|B) = \frac{0.12}{0.12 + 0.28} = 0.3 \)
c) \( P(A') = 0.28 + 0.3 = 0.58 \)，\( P(B \cap A') = 0.28 \)，故 \( P(B|A') = \frac{0.28}{0.58} = \frac{14}{29} \)
d) \( P(B|A \cup B) = \frac{0.12 + 0.28}{0.7} = \frac{4}{7} \)
The Venn diagram shows the probabilities of three events, \( A \), \( B \) and \( C \).
Find:
a) \( P(A|B) \)
b) \( P(C|A') \)
c) \( P((A \cap B)|C') \)
d) \( P(C|(A' \cup B')) \)
(Venn diagram: Only \( A = 0.2 \), \( A \cap B \) (exclusive) = 0.2, Only \( B = 0.12 \), \( A \cap C \) (exclusive) = 0.05, \( B \cap C \) (exclusive) = 0.08, Only \( C = 0.1 \), Neither = 0.15, \( A \cap B \cap C = 0.1 \))
解答过程：
a) \( P(A \cap B) = 0.2 + 0.1 = 0.3 \)，\( P(B) = 0.2 + 0.1 + 0.08 + 0.12 = 0.5 \)，故 \( P(A|B) = \frac{0.3}{0.5} = 0.6 \)
b) \( P(A') = 1 - (0.2 + 0.1 + 0.05) = 0.65 \)，\( P(C \cap A') = 0.08 + 0.1 + 0.15 = 0.33 \)，故 \( P(C|A') = \frac{0.33}{0.65} = \frac{33}{65} \)
c) \( P(C') = 1 - (0.05 + 0.1 + 0.08 + 0.1) = 0.67 \)，\( P((A \cap B) \cap C') = 0.2 \)，故 \( P((A \cap B)|C') = \frac{0.2}{0.67} = \frac{20}{67} \)
d) \( P(A' \cup B') = 1 - P(A \cap B) = 1 - 0.3 = 0.7 \)，\( P(C \cap (A' \cup B')) = 0.05 + 0.08 + 0.15 = 0.28 \)，故 \( P(C|(A' \cup B')) = \frac{0.28}{0.7} = 0.4 \)
The Venn diagram shows the probabilities for events \( A \) and \( B \). Given that \( P(A|B) = P(A') \), find the values of \( c \) and \( d \).
(Venn diagram: Only \( A = 0.3 \), \( A \cap B = c \), Only \( B = d \), Neither = 0.2)
解答过程：
步骤1：计算概率表达式
\( P(A) = 0.3 + c \)，\( P(B) = c + d \)，\( P(A \cap B) = c \)，\( P(A') = 1 - (0.3 + c) = 0.7 - c \)
步骤2：应用条件
给定 \( P(A|B) = P(A') \)，故 \( \frac{c}{c + d} = 0.7 - c \)
步骤3：利用总概率为1
\( 0.3 + c + d + 0.2 = 1 \implies c + d = 0.5 \)
步骤4：代入求解
代入 \( c + d = 0.5 \) 到 \( \frac{c}{0.5} = 0.7 - c \)
\( 2c = 0.7 - c \implies 3c = 0.7 \implies c = \frac{7}{30} \)
故 \( d = 0.5 - \frac{7}{30} = \frac{4}{15} \)
假设维恩图显示：仅A区域0.25，A∩B区域0.15，仅B区域0.20，既不区域0.40。
计算：
a) \( P(A|B) \)
b) \( P(B|A') \)
c) \( P((A \cup B)|A) \)
d) \( P(A'|(A \cup B)') \)
解答过程：
a) \( P(A|B) = \frac{0.15}{0.15 + 0.20} = \frac{0.15}{0.35} = \frac{3}{7} \)
b) \( P(A') = 0.20 + 0.40 = 0.60 \)，\( P(B \cap A') = 0.20 \)，故 \( P(B|A') = \frac{0.20}{0.60} = \frac{1}{3} \)
c) \( P(A \cup B) = 0.25 + 0.15 + 0.20 = 0.60 \)，故 \( P(A \cup B|A) = \frac{0.60}{0.25 + 0.15} = \frac{0.60}{0.40} = 1.5 \)（不可能，检查数据）
d) \( P((A \cup B)') = 0.40 \)，故 \( P(A'|(A \cup B)') = \frac{0.40}{0.40} = 1 \)