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。
概率与条件概率基础
At a movie theater, there are a total of 350 customers. Each customer is located in either theater A, theater B, or theater C. If one of these customers is selected at random, the probability of selecting a customer who is located in theater A is 0.48, and the probability of selecting a customer who is located in theater B is 0.24. How many customers are located in theater C?
原库参考解析
概率与条件概率基础
\( -11,-9,\ 26 \)
A data set of three numbers is shown. If a number from this data set is selected at random, what is the probability of selecting a positive number?
原库参考解析
概率与条件概率基础
The table summarizes members of a local organization by age and whether they live east or west of the river. If a member of the organization is selected at random, what is the probability that the selected member is at least 40 years old?
原库参考解析
概率与条件概率进阶
The table gives the distribution of votes for a new school mascot and grade level for 80 students.
If one of these students is selected at random, what is the probability of selecting a student whose vote for new mascot was for a lion?
原库参考解析
概率与条件概率挑战
The same 20 contestants, on each of 3 days, answered 5 questions in order to win a prize. Each contestant received 1 point for each correct answer. The number of contestants receiving a given score on each day is shown in the table above.
No contestant received the same score on two different days. If a contestant is selected at random, what is the probability that the selected contestant received a score of 5 on Day 2 or Day 3, given that the contestant received a score of 5 on one of the three days?
原库参考解析
概率与条件概率挑战
Human blood can be classified into four common blood types—A, B, AB, and O. It is also characterized by the presence (+) or absence (-) of the rhesus factor. The table above shows the distribution of blood type and rhesus factor for a group of people. If one of these people who is rhesus negative (-) is chosen at random, the probability that the person has blood type B is \( \frac{1}{9} \) . What is the value of x ?
原库参考解析
概率与条件概率挑战
An alumni association survey asked each high school graduate to select the one activity he or she preferred for the association’s next event. Some of the people responded by phone, and the others responded by email. The table above shows the distribution of preferred activity, in percent, for each response type used. For the survey, the number of email responses was twice the number of phone responses. If a person who preferred a picnic is selected at random, what is the probability that the person responded by email?
原库参考解析
概率与条件概率挑战
On May 10, 2015, there were 83 million Internet subscribers in Nigeria. The major Internet providers were MTN, Globacom, Airtel, Etisalat, and Visafone. By September 30, 2015, the number of Internet subscribers in Nigeria had increased to 97 million. If an Internet subscriber in Nigeria on September 30, 2015, is selected at random, the probability that the person selected was an MTN subscriber is 0.43. There were p million MTN subscribers in Nigeria on September 30, 2015. To the nearest integer, what is the value of p ?
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? (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
Employees working for a customer service line at an electric company recorded all the calls last Monday and noted whether the caller asked for repairs and whether the caller asked about a bill. The results are summarized in the table below.
If a caller last Monday who asked about his or her bill is selected at random, which of the following is closest to the probability that the customer also asked for repairs?
原库参考解析
概率与条件概率基础
There are 20 buttons in a bag: 8 white buttons, 2 orange buttons, and 10 brown buttons. If one of these buttons is selected at random, what is the probability of selecting a white button?
原库参考解析
概率与条件概率挑战
In a bag, there are 7 red, 4 white, 33 blue, and 33 yellow cubes. If one of these cubes is selected at random, what is the probability of selecting a cube that is neither blue nor yellow?
原库参考解析
