维恩图练习题 - 掌握事件关系分析与概率计算
以下是5道综合练习题，涵盖维恩图绘制、事件关系分析和复合事件概率计算等核心内容。
There are 25 students in a tutor group at International College. There are 16 students in the tutor group studying Arabic, 14 studying English, and 6 students studying both English and Arabic.
a) Draw a Venn diagram to represent this information.
b) Find the probability that a randomly chosen student in the tutor group:
i) studies English
ii) studies English and Arabic
iii) studies English but not Arabic
iv) does not study English or Arabic.
解答过程：
a) 绘制维恩图
设"阿拉伯语"为集合 \( A \)，"英语"为集合 \( E \)。
b) 概率计算
i) 学习英语的概率：
\( P(\text{English}) = \frac{8 + 6}{25} = \frac{14}{25} = 0.56 \)
ii) 同时学习英语和阿拉伯语的概率：
\( P(\text{English and Arabic}) = \frac{6}{25} = 0.24 \)
iii) 学习英语但不学习阿拉伯语的概率：
\( P(\text{English but not Arabic}) = \frac{8}{25} = 0.32 \)
iv) 既不学习英语也不学习阿拉伯语的概率：
\( P(\text{neither}) = \frac{1}{25} = 0.04 \)
There are 125 diners in a restaurant who were surveyed to find out if they had ordered garlic bread, pasta or cheesecake:
a) Draw a Venn diagram to represent this information.
b) A diner is chosen at random. Find the probability that the diner ordered:
i) all three items
ii) pasta but not cheesecake and not garlic bread
iii) garlic bread and pasta but not cheesecake
iv) none of these items.
解答过程：
a) 绘制维恩图
设"蒜蓉面包"为 \( G \)，"意大利面"为 \( P \)，"芝士蛋糕"为 \( C \)。
计算各区域人数：
b) 概率计算
i) 点所有三种的概率：
\( P(\text{all three}) = \frac{15}{125} = \frac{3}{25} = 0.12 \)
ii) 仅点意大利面的概率：
\( P(\text{pasta only}) = \frac{10}{125} = \frac{2}{25} = 0.08 \)
iii) 点蒜蓉面包和意大利面但不点芝士蛋糕的概率：
\( P(\text{garlic bread and pasta but not cheesecake}) = \frac{10}{125} = \frac{2}{25} = 0.08 \)
iv) 都不点的概率：
\( P(\text{none}) = \frac{54}{125} = 0.432 \)
A group of 275 people at a music festival were asked if they play guitar, piano or drums:
a) Draw a Venn diagram to represent this information.
b) A festival goer is chosen at random from the group. Find:
i) plays the piano
ii) plays at least two of the instruments
iii) plays exactly one of the instruments
iv) plays none of the instruments.
解答过程：
a) 绘制维恩图
设"吉他"为 \( G \)，"钢琴"为 \( P \)，"鼓"为 \( D \)。
计算各区域人数：
b) 概率计算
i) 演奏钢琴的概率：
\( P(\text{piano}) = \frac{15 + 64 + 9 + 1}{275} = \frac{89}{275} \)
ii) 演奏至少两种乐器的概率：
\( P(\text{at least two}) = \frac{64 + 9 + 29 + 1}{275} = \frac{103}{275} \)
iii) 仅演奏一种乐器的概率：
\( P(\text{exactly one}) = \frac{20 + 15 + 35}{275} = \frac{70}{275} = \frac{14}{55} \)
iv) 不演奏任何乐器的概率：
\( P(\text{none}) = \frac{102}{275} \)
The probability that a child in a school has blue eyes is 0.27 and the probability that the child has black hair is 0.35. The probability that the child will have black hair or blue eyes or both is 0.45. A child is chosen at random from the school. Find the probability that the child has:
a) black hair and blue eyes
b) black hair but not blue eyes
c) neither feature.
解答过程：
设"蓝眼睛"为 \( B \)，"黑头发"为 \( H \)。
已知条件：
使用并集公式：
\( P(B \cup H) = P(B) + P(H) - P(B \cap H) \)
\( 0.45 = 0.27 + 0.35 - P(B \cap H) \)
\( P(B \cap H) = 0.27 + 0.35 - 0.45 = 0.17 \)
a) 黑头发且蓝眼睛的概率：
\( P(\text{black hair and blue eyes}) = P(B \cap H) = 0.17 \)
b) 仅黑头发的概率：
\( P(\text{black hair but not blue eyes}) = P(H) - P(B \cap H) = 0.35 - 0.17 = 0.18 \)
c) 既无黑头发也无蓝眼睛的概率：
\( P(\text{neither}) = 1 - P(B \cup H) = 1 - 0.45 = 0.55 \)
A patient going into a doctor's waiting room reads Hiya magazine with probability 0.6 and Dakor magazine with probability 0.4. The probability that the patient reads either one or both of the magazines is 0.7. Find the probability that the patient reads:
a) both magazines
b) Hiya magazine only.
解答过程：
设"读Hiya杂志"为 \( H \)，"读Dakor杂志"为 \( D \)。
已知条件：
使用并集公式：
\( P(H \cup D) = P(H) + P(D) - P(H \cap D) \)
\( 0.7 = 0.6 + 0.4 - P(H \cap D) \)
\( P(H \cap D) = 0.6 + 0.4 - 0.7 = 0.3 \)
a) 读两种杂志的概率：
\( P(\text{both magazines}) = P(H \cap D) = 0.3 \)
b) 仅读Hiya杂志的概率：
\( P(\text{Hiya only}) = P(H) - P(H \cap D) = 0.6 - 0.3 = 0.3 \)