Probability | 概率 - aleveler.com

📚 Probability | 概率

Probability is a core area of Edexcel A-Level Mathematics that models uncertainty and random phenomena. It provides the language and tools for describing how likely events are to occur, and it underpins statistical inference, probability distributions, and hypothesis testing. In the A-Level specification, you are expected to handle basic probability rules, conditional probability, Venn and tree diagrams, discrete random variables, the binomial distribution, the normal distribution, and tests using probability models.

概率是 Edexcel A-Level 数学的核心内容之一，用于对不确定性与随机现象进行建模。它提供了描述事件发生可能性的语言和工具，并为统计推断、概率分布和假设检验奠定了基础。在 A-Level 考试大纲中，你需要掌握基本概率规则、条件概率、韦恩图与树图、离散随机变量、二项分布、正态分布，以及基于概率模型的检验。

1. Basic Probability and Sample Spaces | 基本概率与样本空间

An experiment is any process that produces an outcome, and the set of all possible outcomes is called the sample space, often denoted by S. An event A is a subset of the sample space. If all outcomes are equally likely, the probability of event A is given by the ratio of the number of favourable outcomes to the total number of outcomes.

试验是产生结果的任何过程，所有可能结果的集合称为样本空间，通常记作 S。事件 A 是样本空间的一个子集。如果所有结果等可能，事件 A 的概率由有利结果数与总结果数之比给出。

P(A) = n(A) ÷ n(S)

The probability of any event always lies between 0 and 1 inclusive. A probability of 0 means the event is impossible, while a probability of 1 means the event is certain. The complement of A, written A’, represents the event that A does not occur, and its probability is P(A’) = 1 – P(A). This rule is especially useful when it is easier to calculate the chance that an event does not happen.

任何事件的概率始终介于 0 和 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。A 的补事件记作 A’，表示 A 不发生的事件，其概率为 P(A’) = 1 – P(A)。当计算事件不发生的概率更容易时，这一规则尤其有用。

2. Mutually Exclusive and Exhaustive Events | 互斥事件与完备事件组

Two events A and B are mutually exclusive if they cannot occur at the same time. In this case, the intersection of A and B is empty, so P(A ∩ B) = 0. The addition law then simplifies to P(A ∪ B) = P(A) + P(B), where A ∪ B means A or B or both.

如果两个事件 A 和 B 不能同时发生，则称它们为互斥事件。此时 A 与 B 的交集为空，因此 P(A ∩ B) = 0。加法法则简化为 P(A ∪ B) = P(A) + P(B)，其中 A ∪ B 表示 A 或 B 或两者都发生。

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

A set of events is exhaustive if their union is the whole sample space, meaning that at least one of them must occur. If events are both mutually exclusive and exhaustive, their probabilities add up to 1. For example, when rolling a fair six-sided die, the outcomes ‘odd’ and ‘even’ are mutually exclusive and exhaustive, so P(odd) + P(even) = 1.

如果一组事件的并集是整个样本空间，即它们中至少有一个必然发生，则称这组事件是完备的。如果事件既互斥又完备，它们的概率之和为 1。例如，抛掷一枚均匀的六面骰子时，结果 ‘奇数’ 和 ‘偶数’ 互斥且完备，因此 P(奇数) + P(偶数) = 1。

3. Conditional Probability and Independence | 条件概率与独立事件

Conditional probability measures the chance of event A occurring given that event B has already occurred. It is written P(A | B) and is calculated by dividing the probability of both events happening by the probability of the condition B, provided P(B) > 0.

条件概率度量在事件 B 已经发生的条件下事件 A 发生的概率。它记作 P(A | B)，计算方法是用两事件同时发生的概率除以条件 B 的概率，前提是 P(B) > 0。

P(A | B) = P(A ∩ B) ÷ P(B)

Rearranging this formula gives the multiplication rule: P(A ∩ B) = P(B) × P(A | B). Two events are independent if the occurrence of one does not affect the probability of the other. For independent events, P(A | B) = P(A) and P(A ∩ B) = P(A) × P(B). You should always check independence using the product rule rather than assuming it.

对该公式变形可以得到乘法法则：P(A ∩ B) = P(B) × P(A | B)。如果两个事件中一个的发生不影响另一个发生的概率，则称它们为独立事件。对于独立事件，有 P(A | B) = P(A) 且 P(A ∩ B) = P(A) × P(B)。你应当始终使用乘积法则检验独立性，而不是直接假设。

4. Venn Diagrams and Set Notation | 韦恩图与集合符号

Venn diagrams show the relationship between events using overlapping circles inside a rectangle that represents the sample space. The intersection A ∩ B is the overlap region, the union A ∪ B is the region covered by both circles, and the complement A’ is everything outside circle A. You should be able to shade and interpret such regions.

韦恩图使用矩形表示样本空间，用矩形内重叠的圆来表示事件之间的关系。交集 A ∩ B 是重叠区域，并集 A ∪ B 是两个圆覆盖的区域，补集 A’ 是圆 A 之外的所有区域。你应能正确涂色并解释这些区域。

For two events, the probability that A occurs but B does not is P(A ∩ B’), often called ‘only A’. It is found by subtracting the intersection from P(A), that is P(A ∩ B’) = P(A) – P(A ∩ B). Similarly, P(B ∩ A’) = P(B) – P(A ∩ B). When you are given a Venn diagram or a table of data, these subtraction rules are essential for completing all regions.

对于两个事件，A 发生而 B 不发生的概率为 P(A ∩ B’)，通常称为 ‘只有 A’。其计算方法是 P(A ∩ B’) = P(A) – P(A ∩ B)。类似地，P(B ∩ A’) = P(B) – P(A ∩ B)。当题目给出韦恩图或数据表时，这些减法规则对于填出所有区域至关重要。

5. Tree Diagrams and Probability Multiplication | 树图与概率乘法

Tree diagrams are used to model two or more sequential events. Each branch is labelled with a probability, and the probabilities on any set of branches from a single point must sum to 1. To find the probability of a complete path, you multiply the probabilities along the branches. To find the probability of an event that can occur through multiple paths, you add the probabilities of the relevant terminal outcomes.

树图用于对两个或多个相继发生的事件进行建模。每条分支都标有概率，从同一点分出的所有分支上的概率之和必须为 1。要求出某条完整路径的概率，需要将沿途各分支的概率相乘。若某个事件可以通过多条路径发生，则将该事件相关末端结果的概率相加。

A common context is selecting items with or without replacement. With replacement, the probabilities on the second stage remain the same as the first; without replacement, the denominators and numerators change. Tree diagrams also help when conditional probabilities are given, because second-stage probabilities are often conditioned on the first outcome.

常见情境是有放回或不放回地抽取物品。有放回时，第二阶段的概率与第一阶段保持相同；不放回时，分母和分子会改变。当题目给出条件概率时，树图也非常有用，因为第二阶段的概率通常以第一阶段的结局为条件。

6. Discrete Random Variables and Probability Distributions | 离散随机变量与概率分布

A discrete random variable X takes a set of countable values, each with a certain probability. The probability distribution lists all possible values x and their probabilities P(X = x). For a valid distribution, every probability must be between 0 and 1, and the sum of all probabilities must equal 1.

离散随机变量 X 取一组可数的值，每个值都有一定的概率。概率分布列出所有可能的值 x 及其概率 P(X = x)。对于一个有效的分布，每个概率必须在 0 和 1 之间，且所有概率之和必须等于 1。

Σ P(X = x) = 1

You may also meet the cumulative distribution function F(x) = P(X ≤ x), which gives the probability that X is less than or equal to a particular value. Exam questions often require you to use a probability distribution table to find unknown probabilities, expected values, or probabilities of compound events such as P(X > 2) or P(1 ≤ X < 4).

你可能还会遇到累积分布函数 F(x) = P(X ≤ x)，它表示 X 小于或等于某个特定值的概率。考试题通常要求你利用概率分布表求出未知概率、期望值，或复合事件的概率，如 P(X > 2) 或 P(1 ≤ X < 4)。

7. Expectation and Variance | 期望与方差

The expected value E(X) of a discrete random variable is the long-run average of its values, weighted by their probabilities. It is calculated by summing x times P(X = x) over all possible values. The variance Var(X) measures the spread of the distribution, and the standard deviation is its positive square root.

离散随机变量的期望值 E(X) 是其取值按概率加权的长期平均值。计算方法是将所有可能的 x 与 P(X = x) 的乘积求和。方差 Var(X) 度量分布的离散程度，标准差是方差的正平方根。

E(X) = Σ x P(X = x) | Var(X) = E(X²) – [E(X)]²

Two linear transformation rules are particularly important. For constants a and b, E(aX + b) = aE(X) + b, and Var(aX + b) = a²Var(X). Notice that adding a constant shifts the mean but does not change the variance, while multiplying by a constant scales both the mean and the variance by appropriate factors.

两条线性变换规则特别重要。对于常数 a 和 b，有 E(aX + b) = aE(X) + b，以及 Var(aX + b) = a²Var(X)。注意，加上一个常数会平移均值但不会改变方差，而乘以一个常数则分别以适当的因子缩放均值和方差。

8. Binomial Distribution | 二项分布

The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure. The probability of success p remains constant across all trials. If X is the number of successes in n trials, we write X ~ B(n, p).

二项分布用于建模固定次数的独立试验中成功的次数，其中每次试验只有两种可能结果：成功或失败。成功概率 p 在所有试验中保持不变。如果 X 是 n 次试验中成功的次数，我们记作 X ~ B(n, p)。

P(X = r) = C(n, r) × p^r × (1 – p)^{n – r}

The mean and variance of a binomial random variable are E(X) = np and Var(X) = np(1 – p). These formulas are useful for checking whether a given binomial model is appropriate. In exams, you are expected to use a calculator or statistical tables to find binomial probabilities, especially for cumulative probabilities such as P(X ≤ k) or P(X ≥ k).

二项随机变量的均值和方差分别为 E(X) = np 和 Var(X) = np(1 – p)。这些公式有助于检验给定的二项模型是否合适。在考试中，你需要使用计算器或统计表求二项概率，尤其是累积概率，如 P(X ≤ k) 或 P(X ≥ k)。

9. Normal Distribution as a Probability Model | 正态分布作为概率模型

The normal distribution is a continuous probability distribution used to model many naturally occurring measurements. It is symmetrical and bell-shaped, with mean μ and variance σ², written X ~ N(μ, σ²). The total area under the curve is 1, and probabilities are represented by areas under the curve.

正态分布是一种连续概率分布，用于对许多自然产生的测量值进行建模。它是对称的钟形曲线，均值为 μ，方差为 σ²，记作 X ~ N(μ, σ²)。曲线下的总面积为 1，概率由曲线下的面积表示。

To calculate normal probabilities, you first standardise the variable using Z = (X – μ) ÷ σ, where Z follows the standard normal distribution N(0, 1). Then use tables or a calculator to find P(Z < z). The symmetry of the curve gives P(Z < -z) = P(Z > z) = 1 – P(Z < z). You should also be able to use inverse normal calculations to find unknown means or standard deviations.

要计算正态概率，首先使用 Z = (X – μ) ÷ σ 将变量标准化，其中 Z 服从标准正态分布 N(0, 1)。然后使用表格或计算器求 P(Z < z)。曲线的对称性给出 P(Z < -z) = P(Z > z) = 1 – P(Z < z)。你还应能使用逆正态计算求出未知的均值或标准差。

10. Hypothesis Testing with Probability | 概率背景下的假设检验

Hypothesis testing uses probability to decide whether observed data provide enough evidence to reject a null hypothesis H₀. The alternative hypothesis H₁ states the claim you are testing for. For binomial tests, the test statistic is the number of successes, and the p-value is the probability of obtaining a result at least as extreme as the one observed, assuming H₀ is true.

假设检验使用概率来判断观测数据是否提供了足够证据来拒绝原假设 H₀。备择假设 H₁ 表示你正在检验的主张。对于二项检验，检验统计量是成功的次数，p 值是在 H₀ 为真的条件下，得到至少与观测结果同样极端的结果的概率。

The significance level, usually 5% or 1%, is the threshold for rejecting H₀. If the p-value is less than the significance level, you reject H₀ in favour of H₁; otherwise you do not reject H₀. You must state your conclusion in the context of the original problem. One-tailed tests consider deviations in one direction only, while two-tailed tests consider both directions, so the significance level is split between the two tails.

显著性水平，通常为 5% 或 1%，是拒绝 H₀ 的临界值。如果 p 值小于显著性水平，则拒绝 H₀ 而支持 H₁；否则不拒绝 H₀。你必须结合原始问题的背景陈述结论。单尾检验只考虑一个方向的偏离，而双尾检验考虑两个方向，因此显著性水平要在两个尾部之间分配。

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Probability | 概率