这部分需要会什么？
- 从等可能结果或频数表计算概率
- 正确确定条件概率的分母
- 区分互斥、独立和重复计算
核心概念与公式
等可能情形下，概率为有利结果数除以总结果数。条件概率 \(P(A\mid B)=P(A\cap B)/P(B)\)，要求 \(P(B)>0\)；“已知 B”意味着只在 B 发生的结果中比较。
并集公式为 \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)。独立事件满足 \(P(A\cap B)=P(A)P(B)\)；互斥事件不能同时发生。两个概率都大于零的互斥事件不会独立。
做题时怎样使用
- 确定抽取规则与样本空间，检查结果是否等可能
- 出现“given”“among”等条件时，先缩小分母范围
- 涉及“或”时扣除重复部分，涉及“且”时检查是否真的独立
容易混淆的地方
从表中求“在女生中选择某课程的比例”，分母是女生人数，不是全体人数。不放回抽样一般会改变后续概率。
用一个例题串起方法
知识点教学例题共有 60 人，其中女生 36 人；女生中有 15 人参加社团。从女生中随机选一人，该人参加社团的概率是多少？
展开解题思路 ＋
条件已限定为女生，所以概率为 \(15/36=5/12\)，不能使用 \(15/60\)。
本知识点题库
先独立作答，再查看答案与解析。
概率与条件概率进阶
A total of 50 people attended a conference and were offered 4 types of entrees. The table shows the number of people who chose each type of entree.
If one of these people is selected at random, what is the probability of selecting a person who chose a vegetarian entree?
原库参考解析
概率与条件概率挑战
For an exercise physiology study, 75 participants were assigned to an activity, and their average heart rate while engaged in the activity was recorded. The table summarizes the results of this study.
One of these participants will be selected at random. What is the probability of selecting a participant who was assigned to the swimming activity given that the participant had an average heart rate of less than 130 beats per minute while engaged in an activity? (Express your answer as a decimal or fraction, not as a percent.)
原库参考解析
概率与条件概率挑战
The table shows the distribution of two types of trees at two different sites. If a tree represented in the table is selected at random, what is the probability of selecting tree from site A, given that the tree is a red maple? (Express your answer as a decimal or fraction, not as a percent.)
原库参考解析
概率与条件概率进阶
A bin contains a mixture of T-shirts for two sports teams. The table shows the number of T-shirts in the bin, classified by size and sports team.
One T-shirt from the bin will be selected at random. What is the probability of selecting a T-shirt that is medium, given that the T-shirt is a Flames T-shirt? (Express your answer as a decimal or fraction, not as a percent.)
原库参考解析
概率与条件概率进阶
Each of 181 flowers can be classified as one of three colors, as shown in the frequency table.
If one of the flowers is selected at random, what is the probability of selecting a flower of color Y?
原库参考解析
概率与条件概率进阶
Each of 174 power tools can be classified as one of three types, as shown in the frequency table.
If one of the power tools is selected at random, what is the prohability of selecting a power tool of type Y?
原库参考解析
概率与条件概率进阶
A sleep study consisted of 77 participants, of which 69 participants each had an average of more than 120 minutes of rapid eye movement (REM) sleep per night. If a participant from this sleep study is selected at random, what is the probability of selecting a participant that had an average of more than 120 minutes of REM sleep per night?
原库参考解析
概率与条件概率进阶
A bin contains a mixture of T-shirts for two sports teams. The table shows the number of T-shirts in the bin, classified by size and sports team.
One T-shirt from the bin will be selected at random. What is the probability of selecting a T-shirt that is medium, given that the T-shirt is a Flames T-shirt?
原库参考解析
概率与条件概率进阶
The table summarizes members of a local organization by age and whether they live north or south of Center St. If a member of the organization is selected at random, what is the probability that the selected member is at least 45 years old?
原库参考解析
概率与条件概率挑战
The given table shows the number of even and odd integers contained in list A and list B. No integer is contained in both lists. What fraction of the even integers represented in the table are in list A?
原库参考解析
概率与条件概率进阶
Each face of a fair 12-sided die is labeled with a number from 1 through 12, with a different number appearing on each face. If the die is rolled one time, what is the probability of rolling a 2?
原库参考解析
概率与条件概率进阶
At a convention center, there are a total of 250 visitors. Each visitor is located in either room A, room B, or room C. If one of these visitors is selected at random, the probability of selecting a visitor who is located in room A is 0.44, and the probability of selecting a visitor who is located in room B is 0.32. How many visitors are located in room C?
原库参考解析
