IGCSE Mathematics: Basic Laws of Probability | IGCSE数学：概率的基本定律

📚 IGCSE Mathematics: Basic Laws of Probability | IGCSE数学：概率的基本定律

Probability is the branch of mathematics that measures how likely an event is to happen. In IGCSE Mathematics, probability questions often involve coins, dice, cards, counters, and real-life situations.

概率是衡量事件发生可能性的数学分支。在IGCSE数学中，概率题常常涉及硬币、骰子、扑克牌、计数器和实际生活情境。

1. What Is Probability? | 什么是概率？

Probability is a number that describes the chance of a particular outcome occurring. It is based on a random experiment, which is a process whose result cannot be predicted with absolute certainty.

概率是描述某个特定结果发生机会大小的数值。它基于随机试验，即无法完全确定结果的过程。

For example, when you toss a fair coin, you know the result will be heads or tails, but you cannot predict the exact outcome before it happens.

例如，当你掷一枚均匀硬币时，你知道结果会是正面或反面，但无法在结果出现之前准确预测。

In IGCSE questions, you should first decide whether the problem involves equal outcomes, replacement, independence, or mutually exclusive events. This helps you choose the correct law.

在IGCSE题目中，你首先要判断问题是否涉及等可能结果、是否放回、事件是否独立或是否互斥。这会帮助你选择正确的定律。

2. The Probability Scale | 概率标度

All probabilities must lie between 0 and 1 inclusive. A probability of 0 means an event is impossible, and a probability of 1 means an event is certain.

所有概率必须位于0到1之间，包括0和1。概率为0表示事件不可能发生，概率为1表示事件必然发生。

0: impossible

0：不可能
1/2: even chance

1/2：机会均等
1: certain

1：必然

For an ordinary fair die, the probability of rolling a 4 is 1/6. The probability of rolling a number greater than 6 is 0 because it is impossible.

对于一枚普通均匀骰子，掷出4的概率是1/6。掷出大于6的数的概率是0，因为这是不可能的。

P(event) = number of favourable outcomes ÷ number of equally likely outcomes

P(事件) = 有利结果数 ÷ 等可能结果总数

This formula is valid only when every outcome in the sample space is equally likely, such as with fair coins, fair dice, or well-shuffled cards.

这个公式只有在样本空间中每个结果都等可能时才能使用，例如均匀硬币、均匀骰子或充分洗过的扑克牌。

3. Sample Space and Events | 样本空间与事件

The sample space is the set of all possible outcomes of an experiment. An event is a subset of the sample space, and it may contain one or more outcomes.

样本空间是试验所有可能结果的集合。事件是样本空间的子集，可能包含一个或多个结果。

For a fair die, the sample space is {1, 2, 3, 4, 5, 6}.

对于一枚均匀骰子，样本空间是 {1, 2, 3, 4, 5, 6}。

If event A is “rolling an even number”, then A = {2, 4, 6} and P(A) = 3/6 = 1/2.

如果事件A是“掷出偶数”，那么 A = {2, 4, 6}，且 P(A) = 3/6 = 1/2。

One important property is that the probabilities of all outcomes in a sample space add up to 1. This is a quick check you can use in exams.

一个重要性质是：样本空间中所有结果的概率之和等于1。这是在考试中可以用来快速检查的方法。

4. The Complement Law | 互补事件定律

The complement of event A is written as A′ and means “not A”. It includes every outcome in the sample space that is not in A.

事件A的补事件写作A′，表示“非A”。它包含样本空间中所有不属于A的结果。

P(A′) = 1 − P(A)

For example, if the probability of rain tomorrow is 0.35, then the probability of no rain is 0.65 because 1 − 0.35 = 0.65.

例如，如果明天下雨的概率是0.35，那么不下雨的概率就是0.65，因为 1 − 0.35 = 0.65。

The complement law is especially useful for questions that say “at least one”. For example, P(at least one head) = 1 − P(no heads).

互补事件定律在“至少一次”类问题中特别有用。例如，P(至少一个正面) = 1 − P(没有正面)。

5. Addition Law for Mutually Exclusive Events | 互斥事件的加法法则

Two events are mutually exclusive if they cannot happen at the same time. For example, rolling a 3 and rolling a 5 on a single die are mutually exclusive because only one number can appear.

如果两个事件不可能同时发生，则它们被称为互斥事件。例如，在一次掷骰子中，掷出3和掷出5是互斥的，因为只能出现一个数字。

P(A or B) = P(A) + P(B)

P(A或B) = P(A) + P(B)

If you roll a fair die, the probability of rolling a 3 or a 5 is 1/6 + 1/6 = 2/6 = 1/3.

如果你掷一枚均匀骰子，掷出3或5的概率是 1/6 + 1/6 = 2/6 = 1/3。

The word “or” in probability usually means that at least one of the events occurs. Before adding probabilities, always check whether the events are mutually exclusive.

概率中的“或”通常表示至少一个事件发生。在把概率相加之前，一定要检查事件是否互斥。

6. Addition Law for Non-Mutually Exclusive Events | 非互斥事件的加法法则

When two events can happen at the same time, they are not mutually exclusive. In that case, adding P(A) and P(B) counts the overlap twice.

当两个事件可以同时发生时，它们不是互斥的。在这种情况下，直接把P(A)和P(B)相加会把重叠部分重复计算。

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Here, A ∪ B means “A or B” and A ∩ B means “A and B”.

这里，A ∪ B 表示“A或B”，A ∩ B 表示“A且B”。

Example: from a standard deck of 52 cards, P(red) = 26/52, P(king) = 4/52, and P(red king) = 2/52. Therefore P(red or king) = 26/52 + 4/52 − 2/52 = 28/52 = 7/13.

例如：从一副52张标准扑克牌中，P(红色) = 26/52，P(K) = 4/52，P(红色K) = 2/52。因此 P(红色或K) = 26/52 + 4/52 − 2/52 = 28/52 = 7/13。

If the events are mutually exclusive, then P(A ∩ B) = 0, so this formula becomes the simpler addition law from Section 5.

如果两个事件互斥，那么 P(A ∩ B) = 0，因此这个公式就会变成第5节中更简单的加法法则。

7. Multiplication Law for Independent Events | 独立事件的乘法法则

Two events are independent if the occurrence of one does not affect the probability of the other. For example, the result of tossing a coin does not affect the result of rolling a die.

如果两个事件中一个事件的发生不影响另一个事件发生的概率，那么这两个事件就是独立的。例如，掷硬币的结果不会影响掷骰子的结果。

P(A and B) = P(A) × P(B)

P(A且B) = P(A) × P(B)

For example, the probability of getting an even number on a fair die and heads on a fair coin is 3/6 × 1/2 = 1/2 × 1/2 = 1/4.

例如，在均匀骰子上掷出偶数并且在均匀硬币上掷出正面的概率是 3/6 × 1/2 = 1/2 × 1/2 = 1/4。

The word “and” in probability often signals that you need to multiply. However, you must first check whether the events are independent.

概率中的“且”通常意味着需要做乘法。但是，你必须先检查事件是否独立。

8. Conditional Probability | 条件概率

When events are not independent, the probability of one event depends on whether another event has already happened. This is called conditional probability.

当事件不独立时，一个事件的概率取决于另一个事件是否已经发生。这被称为条件概率。

P(A|B) = P(A ∩ B) ÷ P(B)

P(A|B) means “the probability of A given that B has already happened”.

P(A|B) 表示“在B已经发生的条件下，A发生的概率”。

A bag contains 3 red and 2 blue counters. Two counters are drawn without replacement. The probability that the first counter is red is 3/5. After one red counter has been removed, only 2 red and 2 blue counters remain, so P(second red | first red) = 2/4 = 1/2.

一个袋子中有3个红色计数器和2个蓝色计数器。不放回地抽取两个计数器。第一次抽到红色的概率是3/5。在移除一个红色计数器后，剩下2个红色和2个蓝色计数器，所以 P(第二次红色 | 第一次红色) = 2/4 = 1/2。

For two dependent events, the general multiplication law is P(A and B) = P(A) × P(B|A).

对于两个不独立事件，一般乘法法则是 P(A且B) = P(A) × P(B|A)。

In the counter example, P(both red) = 3/5 × 2/4 = 6/20 = 3/10.

在计数器例子中，P(两次都是红色) = 3/5 × 2/4

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

IGCSE Mathematics: Basic Laws of Probability | IGCSE数学：概率的基本定律