📚 Basic Probability | 基本概率
Probability is a measure of how likely an event is to occur. It is expressed as a number between 0 and 1, where 0 means impossible and 1 means certain. In IGCSE Edexcel Mathematics, you will learn to calculate probabilities for single events, combined events and use tree diagrams to model sequences of events.
概率是衡量事件发生可能性的一种度量。它用一个介于0和1之间的数字表示，0表示不可能，1表示确定。在IGCSE Edexcel数学中，你将学习计算单个事件、组合事件的概率，并使用树形图来模拟一系列事件。
1. What is Probability? | 什么是概率？
Probability is the likelihood that a particular event will happen. It can be written as a fraction, a decimal or a percentage. For example, a probability of 1/2 means the event is equally likely to happen or not happen.
概率是特定事件发生的可能性。它可以写成分数、小数或百分数。例如，概率为1/2意味着事件发生和不发生的可能性相等。
. If we roll a fair six-sided die, the probability of rolling a 4 is P(rolling a 4) = 1/6.
事件A的概率记为P(A)。如果我们掷一个均匀的六面骰子，掷出4的概率是P(掷出4)=1/6。
2. Sample Space | 样本空间
A sample space is the set of all possible outcomes of an experiment. It is often denoted by the letter S. Listing the sample space helps you see all the possibilities clearly.
样本空间是一个实验所有可能结果的集合。它通常用字母S表示。列出样本空间有助于你清楚地看到所有可能的结果。
Example: When flipping a coin, the sample space is S = {'Heads', 'Tails'}. When rolling a die, S = {1, 2, 3, 4, 5, 6}.
例如：抛一枚硬币时，样本空间S={'正面'，'反面'}。掷一个骰子时，S={1,2,3,4,5,6}。
3. Calculating Probability | 计算概率
outcomes) / (Total number of outcomes). This applies when all outcomes are equally likely.
概率的基本公式是：P(A)=(有利结果的数量)/(总结果的数量)。这适用于所有结果等可能的情况。
6 = 1/2, because there are three favourable outcomes (2,4,6) out of six possible outcomes.
对于一个均匀的骰子，掷出偶数的概率是P(偶数)=3/6=1/2，因为有三种有利的结果(2,4,6)在六种可能的结果中。
4. Probability Scale | 概率尺度
Probabilities range from 0 to 1. An impossible event has probability 0, a certain event has probability 1, and a likely but not certain event has a probability between 0.5 and 1.
概率的范围从0到1。不可能事件的概率为0，必然事件的概率为1，很可能但不一定的事件概率在0.5到1之间。
You should be able to place the probability of everyday events on the probability scale, such as 'it will rain tomorrow' or 'you will win a lottery'.
你应该能够将日常事件的概率放在概率尺度上，例如“明天会下雨”或“你会中彩票”。
5. Mutually Exclusive Events | 互斥事件
Two events are mutually exclusive if they cannot happen at the same time. For example, when rolling a die, the events 'roll an odd number' and 'roll an even number' are mutually exclusive.
如果两个事件不能同时发生，则它们是互斥的。例如，掷骰子时，“掷出奇数”和“掷出偶数”是互斥事件。
For mutually exclusive events A and B, P(A or B) = P(A) + P(B). This is the addition rule.
对于互斥事件A和B，P(A或B)=P(A)+P(B)。这就是加法法则。
6. Independent Events | 独立事件
Events are independent if the outcome of one does not affect the outcome of the other. For example, flipping a coin and rolling a die are independent events.
如果事件的结果互不影响，则它们是独立事件。例如，抛硬币和掷骰子是独立事件。
= P(A) × P(B). This multiplication rule is crucial for solving many probability problems.
对于独立事件A和B，P(A且B)=P(A)×P(B)。这个乘法法则对于解决许多概率问题至关重要。
7. Tree Diagrams | 树形图
Tree diagrams are used to represent sequences of events, especially when events are independent or conditional. Each branch represents a possible outcome, and you write the probability along the branch.
树形图用于表示事件序列，特别是当事件独立或条件相关时。每个分支代表一个可能的结果，你沿着分支写上概率。
To find the probability of a combined event, multiply the probabilities along the branches that lead to that outcome. If more than one path leads to the desired outcome, add the probabilities of those paths.
要找到组合事件的概率，沿着通向该结果的分支将概率相乘。如果有多条路径通向所需的结果，则把这些路径的概率相加。
m a deck with replacement. The probability of drawing two aces is (4/52) × (4/52) = 1/169.
例子：有放回地从一副牌中抽取两张牌。抽出两张A的概率是(4/52)×(4/52)=1/169。
8. Experimental Probability | 实验概率
. It is calculated as: P(E) = (Number of times event E occurs) / (Total number of trials).
实验概率是基于实际试验或实验的。它的计算方法是：P(E)=(事件E发生的次数)/(总试验次数)。
As the number of trials increases, the experimental probability tends to get closer to the theoretical probability. This is known as the Law of Large Numbers.
随着试验次数的增加，实验概率往往接近于理论概率。这被称为大数定律。
9. Expected Frequency | 期望频率
Expected frequency is the number of times you would expect an event to occur in a number of trials. It is found by multiplying the probability of the event by the number of trials.
期望频率是在一定次数的试验中，你预期事件会发生的次数。它通过事件的概率乘以试验次数得到。
you roll a die 300 times, the expected frequency of rolling a 5 is (1/6) × 300 = 50 times.
例如，如果你掷一个骰子300次，掷出5的期望频率是(1/6)×300=50次。
10. Venn Diagrams and Probability | 维恩图与概率
Venn diagrams can be used to show overlapping events. They help in calculating probabilities involving 'and' (intersection) and 'or' (union).
维恩图可以用来表示重叠的事件。它们有助于计算涉及“且”(交集)和“或”(并集)的概率。
The probability of the union of two events is P(A ∪ B) = P(A) + P(B) - P(A ∩ B). This formula avoids double-counting the intersection.
两个事件的并集概率为P(A ∪ B)=P(A)+P(B)-P(A ∩ B)。这个公式避免了重复计算交集。
11. Conditional Probability | 条件概率
of an event occurring given that another event has already occurred. It is written as P(A|B) and calculated as P(A|B) = P(A ∩ B) / P(B).
条件概率是在已知另一个事件已经发生的情况下，某个事件发生的概率。它写作P(A|B)，计算公式为P(A|B)=P(A ∩ B)/P(B)。
In IGCSE, conditional probability questions often involve tree diagrams without replacement. Always adjust the probabilities when the situation changes.
在IGCSE中，条件概率题目通常涉及不放回的树形图。当情况改变时，一定要调整概率。
12. Summary and Key Points | 总结与要点
Remember: probability is always between 0 and 1. Use the addition rule for 'or' with mutually exclusive events, and the multiplication rule for 'and' with independent events. Draw tree diagrams for multi-stage events, and write all probabilities clearly. For conditional probability, think about how the sample space reduces.
记住：概率总是在0和1之间。对于互斥事件的“或”使用加法法则，对于独立事件的“且”使用乘法法则。为多阶段事件画树形图，并清楚地写出所有概率。对于条件概率，思考样本空间如何缩小。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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