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
For a school competition, each student in the sixth, seventh, and eighth grades is assigned to either the red team or the green team. The table shows the distribution of grade and team for the students in the school.
A student from the competition will be selected at random. What is the probability of selecting a student who is in the sixth or seventh grade, given that the student is on the green team? (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
Hakim has a bag of 50 tiles, where 45 are labeled with a letter and 5 are blank. If Hakim selects a tile from the bag at random, what is the probability of selecting a blank tile? (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
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 0.48, 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 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 0.72 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
概率与条件概率进阶
Each rock in a collection of 70 rocks was classified as either igneous, metamorphic, or sedimentary, as shown in the frequency table.
If one of these rocks is selected at random, what is the probability of selecting a rock that is igneous?
原库参考解析
概率与条件概率挑战
The table summarizes the distribution of age and assigned group for 90 participants in a study.
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
概率与条件概率挑战
The table summarizes the distribution of age and assigned group for 120 participants in a study.
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 summarizes the distribution of age and assigned group for 120 participants in a study.
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
概率与条件概率挑战
For 100 neurons, the table summarizes the distribution of classification and cell body diameter.
One of these neurons will be selected at random. What is the probability of selecting a neuron with a cell body diameter that is less than or equal to 30 micrometers, given that it is not classified as a motor neuron? (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
A department store sells pants in three lengths: short, regular, and long. The table summarizes the distribution of pant length, by color, for all the pants in the store.
A pair of pants is selected at random. What is the probability of selecting a pair of pants that is brown, given that the pair of pants is regular length? (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 rock in a collection of 70 rocks was classified as either igneous, metamorphic, or sedimentary, as shown in the frequency table.
If one of these rocks is selected at random, what is the probability of selecting a rock that is igneous?
原库参考解析
概率与条件概率基础
Each of 185 animals can be classified as one of three species, as shown in the frequency table.
If one of the animals is selected at random, what is the probability of selecting an animal of species Y?
原库参考解析
