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。
概率与条件概率挑战
The table shows the distribution of people in a certain city by age group.
If a person in this city is selected at random, which of the following is closest to the probability of selecting a person who is greater than 65 years old, given that the person is at least 18 years old?
原库参考解析
概率与条件概率进阶
A choir of 48 singers has 12 members who sing bass. If one singer from the choir is selected at random, what is the probability of selecting a singer who sings bass?
原库参考解析
概率与条件概率挑战
The table shows the distribution of people in a certain city by age group.
If a person in this city is selected at random, which of the following is closest to the probability of selecting a person who is greater than 65 years old, given that the person is at least 18 years old?
原库参考解析
概率与条件概率挑战
One of these participants will be selected at random. What is the probability of selecting a participant from group A, given that the participant is at least 10 years of age?
原库参考解析
概率与条件概率进阶
The table gives the distribution of flavor and type of topping for customer orders at an ice cream shop.
If a customer order is selected at random, what is the probability of selecting an order with sprinkles, given the flavor of ice cream is vanilla?
原库参考解析
概率与条件概率挑战
At a convention center, there are a total of 225 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 0.72, and the probability of selecting a visitor who is located in room B is 0.24. How many visitors are located in room C?
原库参考解析
概率与条件概率挑战
At a convention center, there are a total of 375 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 0.64, 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
概率与条件概率进阶
At a convention center there are a total of 275 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 0.68, and the probability of selecting a visitor who is located in room B is 0.24. How many visitors are located in room C?
原库参考解析
概率与条件概率挑战
At a convention center, there are a total of 375 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 0.68, and the probability of selecting a visitor who is located in room B is 0.24. How many visitors are located in room C?
原库参考解析
概率与条件概率挑战
At a convention center, there are a total of 325 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 0.64, 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
概率与条件概率进阶
A sleep study consisted of 66 participants, of which 59 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
概率与条件概率挑战
The table shows the distribution of people in a certain city by age group.
If a person in this city is selected at random, which of the following is closest to the probability of selecting a person who is greater than 65 years old, given that the person is at least 18 years old?
原库参考解析
