ESAT Mathematics: Statistics & Probability | ESAT 数学：统计与概率专题

📚 ESAT Mathematics: Statistics & Probability | ESAT 数学：统计与概率专题

The ESAT (Engineering and Science Admissions Test) mathematics section requires a solid grasp of statistics and probability. Candidates must interpret data, calculate probabilities, and apply distributions quickly without relying on calculators. This article compiles essential concepts, worked examples, and common pitfalls for focused revision.

ESAT（工程与科学入学考试）数学部分要求扎实掌握统计与概率。考生必须快速解读数据、计算概率并应用分布，而且通常不能依赖计算器。本文汇总核心概念、典型例题与常见易错点，帮助高效复习。

1. Probability Fundamentals: Events and Sample Space | 概率基础：事件与样本空间

A random experiment has all possible outcomes listed in a sample space S. An event A is a subset of S. The probability of A is the sum of probabilities of the outcomes in A, always between 0 and 1 inclusive.

随机试验的所有可能结果构成样本空间 S。事件 A 是 S 的一个子集。事件 A 的概率等于 A 中所有结果概率之和，取值总在 0 到 1 之间（含端点）。

P(A) = n(A) / n(S) for equally likely outcomes

P(A) = n(A) / n(S)（等可能情形）

For complement events: P(A′) = 1 − P(A). For mutually exclusive events A and B: P(A ∪ B) = P(A) + P(B). For general events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

补事件：P(A′) = 1 − P(A)。若 A、B 互斥，则 P(A ∪ B) = P(A) + P(B)。一般情况下：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。

Identify the sample space first; missing outcomes cause most errors.
先写出样本空间；漏掉结果是最常见的错误来源。

2. Conditional Probability and Independence | 条件概率与独立性

The conditional probability of A given B is written P(A|B) and defined as P(A ∩ B) / P(B), provided P(B) > 0.

事件 B 发生条件下事件 A 的条件概率记为 P(A|B)，定义为 P(A ∩ B) / P(B)，要求 P(B) > 0。

P(A|B) = P(A ∩ B) / P(B)

Events A and B are independent if P(A ∩ B) = P(A) × P(B). Equivalently, P(A|B) = P(A). Independence is not the same as mutual exclusivity.

事件 A 和 B 独立当且仅当 P(A ∩ B) = P(A) × P(B)，等价地 P(A|B) = P(A)。注意“独立”与“互斥”完全不同。

Use tree diagrams for multi-stage events; branch probabilities multiply, and final probabilities add.
多阶段事件使用树形图；每条路径概率相乘，最终概率相加。

3. Bayes’ Theorem | 贝叶斯定理

Bayes’ theorem revises probabilities when new information arrives. For a partition of the sample space into B₁, B₂, …, Bₙ:

贝叶斯定理用于在获得新信息后修正概率。若样本空间被 B₁, B₂, …, Bₙ 分割，则：

P(Bᵢ|A) = P(Bᵢ) P(A|Bᵢ) / Σⱼ P(Bⱼ) P(A|Bⱼ)

In ESAT problems, identify the “prior probabilities” P(Bᵢ) and the “likelihoods” P(A|Bᵢ). The denominator is the total probability of A.

在 ESAT 题目中，要识别“先验概率” P(Bᵢ) 和“似然” P(A|Bᵢ)。分母就是事件 A 的全概率。

Draw a tree diagram and condition on the correct branch.
画树形图，并在正确的分支上取条件。

4. Random Variables and Probability Distributions | 随机变量与概率分布

A discrete random variable X takes countable values x with probabilities P(X = x) = p(x). These probabilities must sum to 1. A continuous random variable has a probability density function (pdf) with total area 1.

离散型随机变量 X 取可数多个值 x，对应概率 P(X = x) = p(x)。所有概率之和必须为 1。连续型随机变量有概率密度函数（pdf），其总面积为 1。

Σ p(x) = 1 ; ∫ f(x) dx = 1

Always check that a given table or function satisfies the total probability requirement before solving further parts.

在继续解题前，务必先检查所给表格或函数是否满足总概率为 1 的条件。

5. Expectation and Variance | 期望与方差

The expected value E(X) is the long-run average: E(X) = Σ x p(x) for discrete X, or ∫ x f(x) dx for continuous X. The variance measures spread: Var(X) = E(X²) − [E(X)]².

期望 E(X) 是长期平均值：离散型 X 有 E(X) = Σ x p(x)，连续型 X 有 E(X) = ∫ x f(x) dx。方差衡量离散程度：Var(X) = E(X²) − [E(X)]²。

Var(X) = Σ (x − μ)² p(x) = E(X²) − μ²

For linear transformations: E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X). This is heavily tested in ESAT.

线性变换：E(aX + b) = aE(X) + b，Var(aX + b) = a²Var(X)。这是 ESAT 的高频考点。

6. Binomial Distribution | 二项分布

If X counts successes in n independent trials, each with success probability p, then X ~ B(n, p): P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ.

若 X 表示 n 次独立重复试验中的成功次数，每次成功概率为 p，则 X ~ B(n, p)：P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ。

E(X) = np ; Var(X) = np(1−p)

Conditions: fixed n, two outcomes per trial, constant p, independent trials.
条件：固定试验次数 n；每次只有两种结果；成功概率 p 恒定；各次独立。
Use the binomial formula only when all four conditions are met.
只有四个条件均满足时才可使用二项分布公式。

7. Normal Distribution and Normal Approximation | 正态分布与正态近似

A continuous random variable X follows N(μ, σ²) if its pdf is the symmetric bell-shaped curve. Standardisation converts any normal distribution to Z ~ N(0,1):

若连续随机变量 X 的概率密度曲线是钟形对称的，则 X ~ N(μ, σ²)。标准化可将任意正态分布转化为 Z ~ N(0,1)：

Z = (X − μ) / σ

For binomial approximation, use np and np(1−p); apply continuity correction when approximating discrete counts. Common rule: both np and n(1−p) ≥ 5.

二项分布的正态近似使用 μ = np，σ² = np(1−p)；用连续型近似离散计数时必须作连续性修正。常用规则：np 与 n(1−p) 均不小于 5。

Know key z-values: 1.96 for 95%, 1.645 for 90%, 2.576 for 99%.
熟记常用 z 值：95% 对应 1.96，90% 对应 1.645，99% 对应 2.576。
Always draw the normal curve and shade the required region.
解题时先画出正态曲线并标出所求区域。

8. Sampling and Estimation | 抽样与估计

The sample mean x̄ estimates the population mean μ. For a sample of size n from N(μ, σ²), the sampling distribution of x̄ is N(μ, σ²/n). The standard error is σ/√n.

样本均值 x̄ 用于估计总体均值 μ。若样本来自 N(μ, σ²)，则样本均值 x̄ 的抽样分布为 N(μ, σ²/n)。标准误为 σ/√n。

SE(x̄) = σ / √n

A confidence interval for μ when σ is known: x̄ ± z* × σ/√n. ESAT may ask for the interpretation rather than the calculation.

当 σ 已知时，μ 的置信区间为 x̄ ± z* × σ/√n。ESAT 可能更注重对区间含义的理解而不是纯计算。

Larger samples produce narrower intervals.
样本量越大，置信区间越窄。

9. Correlation and Regression | 相关与回归

Pearson’s correlation coefficient r measures linear association between two variables. It lies between −1 and 1. Regression lines are used to predict one variable from another.

皮尔逊相关系数 r 衡量两个变量间的线性相关程度，取值范围在 −1 到 1 之间。回归直线用于由一个变量预测另一个变量。

r = Σ(xᵢ − x̄)(yᵢ − ȳ) / √[Σ(xᵢ − x̄)² Σ(yᵢ − ȳ)²]

Correlation does not imply causation.
相关不等于因果。
Regression predictions are only reliable within the range of observed x-values.
回归预测仅在观测 x 值的范围内可信。

10. Key Strategies and Common Pitfalls | 关键策略与常见易错点

Strategy 1: Use complements for “at least one” probabilities. P(at least one) = 1 − P(none).

策略一：求“至少一个”的概率时用补事件：P(至少一个) = 1 − P(一个也没有)。

Strategy 2: In conditional probability questions, label events clearly and rewrite the question in symbols.

策略二：在条件概率题中，把事件清晰命名，并用符号重写问题。

Strategy 3: For binomial and normal calculations, check the parameters before substituting.

策略三：对二项分布和正态分布，代入前先核对参数。

Pitfall: Confusing P(A|B) with P(B|A). Bayes’ theorem is the bridge.

易错点：混淆 P(A|B) 和 P(B|A)。两者通过贝叶斯定理联系。

Pitfall: Forgetting the continuity correction when approximating a discrete distribution with a continuous one.

易错点：用连续分布近似离散分布时忘记连续性修正。

Pitfall: Assuming equal probability when outcomes are not equally likely, especially in data-based questions.

易错点：在数据题中默认所有结果等可能，而实际上并非如此。

Mastering these core topics — probability rules, conditional probability, distributions, expectation, and sampling — will directly boost your ESAT mathematics score. Practise interpreting word problems quickly and express every probability in set notation before calculating.

掌握以上核心内容——概率规则、条件概率、分布、期望与抽样——将直接提高你的 ESAT 数学成绩。练习快速理解应用题，并在计算前先用集合符号表达每个概率。

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ESAT Mathematics: Statistics & Probability | ESAT 数学：统计与概率专题