Mastering IB Mathematics HL Option: Statistics and Probability | 掌握 IB 数学 HL 选修：统计与概率

📚 Mastering IB Mathematics HL Option: Statistics and Probability | 掌握 IB 数学 HL 选修：统计与概率

The IB Mathematics HL Option: Statistics and Probability is a demanding but rewarding topic that extends core probability ideas into formal inference, distributions and statistical modelling. Success depends on mastering notation, understanding the assumptions behind each model, and using the graphical display calculator (GDC) efficiently to compute probabilities, confidence intervals and p-values.

IB 数学 HL 选修：统计与概率是一门要求高但回报丰厚的专题，它将核心概率思想扩展到形式化推断、分布和统计建模。成功的关键在于掌握符号、理解每个模型背后的假设，并高效使用图形计算器（GDC）来计算概率、置信区间和 p 值。

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1. Axioms of Probability and Set Notation | 概率公理与集合符号

≤ 1; the sample space S satisfies P(S) = 1; and for mutually exclusive events A and B, P(A ∪ ) + P(B). Every other rule you meet in this option is derived from these three statements.

每一个概率模型都建立在三条公理之上：对任意事件 A，0 ≤ P(A) ≤ 1；样本空间 S 满足 P(S) = 1；对于互斥事件 A 和 B，有 P(A ∪ B) = P(A) + P(B)。本选修中你遇到的其他所有规则都由这三条推出。

Inclusion-exclusion extends the addition rule to overlapping events:

容斥原理将加法规则推广到重叠事件：

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

The complement rule is often faster when a question says 'at least one':

当题目出现 "至少一个" 时，互补事件规则通常更快：

P(A') = 1 − P(A)

ccurrence of one does not change the probability of the other. Formally, A and B are independent if and only if P(A ∩ ndence with mutual exclusivity: mutually exclusive events with non-zero probabilities are actually strongly dependent.

当一件事的发生不改变另一件事的概率时，两个事件相互独立。形式上，A 与 B 独立的充要条件是 P(A ∩ B) = P(A) × P(B)。不要把独立与互斥混为一谈：概率非零的互斥事件实际上是强烈相关的。

Drawing a Venn diagram or a two-way table usually clarifies the required region, especially when a question mixes notation such as A ∩ B', A' ∪ B or 'exactly one of A and B'.

画维恩图或双向表通常能厘清所需区域，尤其当题目混合使用 A ∩ B'、A' ∪ B 或 "A 与 B 中恰好一个" 这类符号时。

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2. Conditional Probability and Bayes' Theorem | 条件概率与贝叶斯定理

Conditional probability measures the chance of A occurring given that B has already occurred:

条件概率衡量在 B 已经发生的情况下 A 发生的概率：

P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0

Rearranging this definition gives the multiplication rule, P(A ∩ B) = P(B) × P(A| h is the workhorse of multi-stage problems. For three events the chain rule extends to P(A ∩ B ∩ C) = P(A) × P(B|A) × P(C|A ∩ B).

重排该定义即得乘法法则 P(A ∩ B) = P(B) × P(A|B)，它是多阶段问题的核心工具。对三个事件，链式法则可扩展为 P(A ∩ B ∩ C) = P(A) × P(B|A) × P(C|A ∩ B)。

Bayes' theorem reverses the conditioning. In its extended form it links prior probabilities, likelihoods and posterior probabilities:

贝叶斯定理反转条件关系。其扩展形式将先验概率、似然和后验概率联系起来：

P(A|B) = P(B|A) P(A) / [ P(B|A) P(A) + P(B|A') P(A') ]

The denominator is simply P(B), computed by the law of total probability over the two complementary cases A and A'. This structure explains why Bayes' theorem appears so often in medical screening questions.

分母其实就是 P(B)，通过对互补情形 A 与 A' 使用全概率公式计算得到。正是这一结构解释了为什么贝叶斯定理在医学筛查题中如此常见。

Tree diagrams are strongly recommended for multi-stage problems, especially when distinguishing false positives and false negatives in diagnostic testing. Label the first set of branches with the prior probabilities (for example, 'has disease' and 'does not have disease'), then the second set with the conditional probabilities of the test result. Multiplying along a branch gives a joint probability; adding the relevant branches gives the denominator of Bayes' theorem.

对于多阶段问题，强烈建议使用树状图，尤其是在诊断检测中区分假阳性和假阴性时。第一层分支标注先验概率（例如 "患病" 与 "未患病"），第二层分支标注检测结果的条件概率。沿分支相乘得到联合概率；把相关分支相加即得贝叶斯定理的分母。

A classic exam trap: a highly accurate test can still produce a large proportion of false positives when the disease is rare, because the posterior probability depends on the prevalence as well as the sensitivity and specificity.

一个经典的考试陷阱：即使检测非常准确，当疾病本身罕见时仍可能产生大量假阳性，因为后验概率同时取决于患病率、灵敏度和特异度。

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3. Discrete Random Variables and Expectation | 离散随机变量与期望

. The values x must satisfy P(X = x) ≥ 0 and Σ P(X = x) = 1 over all possible values of x.

离散随机变量 X 由其概率质量函数 P(X = x) 定义。所有可能取值 x 必须满足 P(X = x) ≥ 0 且 Σ P(X = x) = 1。

Its expected value is the long-run average, obtained by weighting each value by its probability:

它的期望值是长期平均，通过将每个取值按其概率加权得到：

E(X) = Σ x P(X = x)

The expectation of a function of X is computed the same way, replacing x with g(x):

X 的函数的期望用同样方式计算，只需把 x 换成 g(x)：

E(g(X)) = Σ g(x) P(X = x)

Variance measures spread around the mean. The computational form is almost always quicker than the definition:

方差衡量围绕均值的离散程度。计算式几乎总比定义式更快：

Var(X) = E(X²) − [E(X)]²

. A useful first step in any exam question is to check whether the distribution is symmetric; if it is, the mean equals the median and you can often read E(X) straight off the table.

标准差为 σ = √Var(X)。考试中一个有用的第一步是检查分布是否对称；如果对称，均值等于中位数，你往往能直接从表格读出 E(X)。

On the GDC you can enter the values in one list and the probabilities in a second list, then use one-variable statistics with the probability list as the frequency list to obtain both the mean and the standard deviation without manual arithmetic.

在 GDC 上，你可以把一个列表填入取值、另一个列表填入概率，然后以一变量统计并把概率列表作为频数列表，即可直接得到均值与标准差，无需手算。

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4. Linear Transformations and Standardisation | 线性变换与标准化

If X is a random variable and a, b are constants, expectation is linear:

若 X 是随机变量，a、b 为常数，则期望是线性的：

E(aX + b) = a E(X) + b

Variance, however, ignores the shift and squares the scaling factor, because shifting every value by b does not change the spread:

但方差忽略平移并对缩放因子取平方，因为将每个取值平移 b 并不改变离散程度：

Var(aX + b) = a² Var(X)

Therefore the standard deviation transforms as SD(aX + b) = |a| e matters: a negative multiplier flips the distribution but never makes a spread negative.

因此标准差按 SD(aX + b) = |a| SD(X) 变换。绝对值很关键：负的乘数会翻转分布，但绝不会让离散程度变成负值。

Standardisation is the most important special case. Subtracting the mean and dividing by the standard deviation always produces a variable with mean 0 and variance 1:

标准化是最重要的特例。减去均值再除以标准差，总是产生均值为 0、方差为 1 的变量：

Z = (X − μ) / σ, so E(Z) = 0 and Var(Z) = 1

This is precisely the transformation used for the normal distribution, and it is also the reason z-scores are comparable across different scales and units.

这正是正态分布所使用的变换，也是 z 分数可以在不同量纲与单位之间互相比较的原因。

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5. The Binomial Distribution | 二项分布

The binomial distribution models the number of successes in a fixed number n of independent trials, each with the same probability of success p. Four conditions must hold: a fixed number of trials, two possible outcomes per trial, a constant probability of success, and independence between trials.

二项分布描述在固定次数 n 的独立试验中成功的次数，每次试验成功概率相同为 p。必须满足四个条件：试验次数固定、每次试验只有两种结果、成功概率恒定、各次试验相互独立。

The probability mass function is:

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其概率质量函数为：

P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ, r = 0, 1, 2, …, n

The mean and variance follow directly from the n independent Bernoulli trials being added together:

由于二项变量是 n 个独立伯努利变量之和，其均值与方差可直接得到：

E(X) = np,   Var(X) = np(1 − p)

topic come from using the wrong one, or from forgetting that P(X ≥ r) = 1 − P(X ≤ r − 1).

在 GDC 上，binompdf(n, p, r) 给出精确概率 P(X = r)，而 binomcdf(n, p, r) 给出 P(X ≤ r)。本专题中的多数错误来自用错函数，或忘记 P(X ≥ r) = 1 − P(X ≤ r − 1)。

Be careful with wording. 'At least three' means P(X ≥ 3). 'More than three' means P(X ≥ 4). 'No more than three' means P(X ≤ 3). Underline these phrases before you press any key.

注意措辞。"至少三个" 指 P(X ≥ 3)。"多于三个" 指 P(X ≥ 4)。"不超过三个" 指 P(X ≤ 3)。按键之前先把这些短语划出来。

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6. The Poisson Distribution | 泊松分布

The Poisson distribution models the number of occurrences of a random event in a fixed interval of time, length, area or volume, when occurrences happen independently and at a constant average rate.

泊松分布描述在固定时间、长度、面积或体积区间内随机事件发生的次数，前提是事件独立发生且平均发生率恒定。

Its probability mass function is:

其概率质量函数为：

P(X = r) = e⁻ᵐ mʳ / r!, r = 0, 1, 2, …

The defining feature of the Poisson distribution is that the mean and the variance are equal:

泊松分布的决定性特征是均值与方差相等：

E(X) = Var(X) = m

nterval changes, the parameter changes proportionally: events at a rate of 3 per hour become m = 1.5 over half an hour.

如果题目给出的均值与方差明显不同，该分布几乎肯定是二项分布而非泊松分布。若区间改变，参数按比例变化：每小时 3 次的事件在半小时内变为 m = 1.5。

is n ≥ 50 and p ≤ 0.05, though the IB mark scheme usually states the condition explicitly.

当 n 大而 p 小时，泊松分布可用作二项分布的近似，取 m = np。常用经验法则是 n ≥ 50 且 p ≤ 0.05，不过 IB 评分方案通常会明确给出该条件。

Feature / 特征	Binomial / 二项	Poisson / 泊松
Parameter / 参数	n, p	m
Mean / 均值	np	m
Variance / 方差	np(1 − p)	m
Upper bound / 上界	n (finite) / 有限	none / 无
GDC command / GDC 命令	binompdf, binomcdf	poissonpdf, poissoncdf

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7. Continuous Random Variables and Probability Density Functions | 连续随机变量与概率密度函数

continuous variable, P(X = a) = 0 for every single value a. Probabilities are only defined over intervals, and they are given by areas under the curve.

连续随机变量由概率密度函数 f(x) 而非概率质量函数描述。核心要点是：对连续变量，任何单一取值的概率 P(X = a) = 0。概率只在区间上定义，并由曲线下的面积给出。

The density must be non-negative everywhere and must integrate to one over its support:

密度函数处处非负，且在其支撑集上积分等于 1：

∫ f(x) dx = 1   (over the whole support)

The probability of an interval is the definite integral of the density over that interval:

某区间的概率是该密度函数在该区间上的定积分：

P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx

The mean and variance are defined by integrals in the same way that sums define them for discrete variables:

均值与方差以积分定义，正如离散变量以求和定义一样：

E(X) = ∫ x f(x) dx,   Var(X) = E(X²) − [E(X)]²

X < b). This is not true for discrete variables, and mixing the two cases up is one of the most frequent mistakes in this option.

由于 P(X = a) = 0，对连续变量而言不等号 ≤ 与 < 给出相同答案，故 P(X ≤ b) = P(X < b)。对离散变量这并不成立，混淆这两种情形是本选修中最常见的错误之一。

ode is the value where the density is maximal. Questions often ask you to find an unknown constant k in a piecewise density by setting the total integral equal to 1.

连续分布的中位数 m 满足 P(X ≤ m) = 0.5，众数是密度取最大值的点。题目常要求你通过令总积分等于 1 来求分段密度函数中的未知常数 k。

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8. The Normal Distribution and Standardisation | 正态分布与标准化

nuous model in the option. It is fully determined by its mean μ and variance σ², written X ~ N(μ, σ²), and it is symmetric about x = μ with a total area of 1 under the curve.

正态分布是本选修中最重要的连续模型。它完全由均值 μ 与方差 σ² 决定，记作 X ~ N(μ, σ²)，关于 x = μ 对称，曲线下总面积为 1。

To use tables or a GDC, convert every value to a z-score:

使用数表或 GDC 时，把每个取值转换为 z 分数：

Z = (X − μ) / σ, where Z ~ N(0, 1)

.96 < Z < 1.96) ≈ 0.95. These four values reappear constantly in confidence intervals and hypothesis tests.

应当立刻认出的标准正态概率包括 P(Z < 1.96) ≈ 0.975、P(Z < 1.645) ≈ 0.95、P(Z < 2.576) ≈ 0.995 以及 P(−1.96 < Z < 1.96) ≈ 0.95。这四个值在置信区间与假设检验中反复出现。

On the GDC, normcdf(lower, upper, μ, σ) gives the area between two values, and invNorm(area, μ, σ) gives the value below which a specified proportion of the distribution lies. Questions that give you a probability and ask for a mark or a cut-off score are always inverse normal problems.

在 GDC 上，normcdf(lower, upper, μ, σ) 给出两个值之间的面积，invNorm(area, μ, σ) 给出低于某个指定比例的分界点。凡是给出概率而要求分数或临界值的题目，都是逆正态问题。

k sketch with the mean marked at the centre prevents sign errors and helps you sanity-check whether your answer should be greater or less than 0.5.

计算之前先利用对称性：P(X > a) = 1 − P(X < a)，且 P(X < μ − d) = P(X > μ + d)。快速画图并在中心标出均值，可避免符号错误，并帮助你检查答案应大于还是小于 0.5。

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9. Linear Combinations of Independent Random Variables | 独立随机变量的线性组合

Expectation is always additive, whether or not the variables are independent:

无论变量是否独立，期望总是可加的：

E(X ± Y) = E(X) ± E(Y)

Variance, by contrast, is additive only when the variables are independent, and crucially the variances add even for a difference:

相比之下，方差只有在变量独立时才可加，而且关键在于即使是差，方差也是相加：

Var(X ± Y) = Var(X) + Var(Y)   (X and Y independent)

er than removing it. A frequent exam error is writing Var(X − Y) = Var(X) − Var(Y), which can even give a negative number.

最后这个结果让许多学生惊讶：X − Y 的方差比单独的 X 更大，因为减去一个不确定量会增加而非减少不确定性。常见考试错误是写成 Var(X − Y) = Var(X) − Var(Y)，这样甚至可能得到负数。

Combining the two rules with scalar multipliers gives the general formula:

把两条规则与标量乘数结合，得到一般公式：

E(aX + bY) = a E(X) + b E(Y)

Var(aX + bY) = a² Var(X) + b² Var(Y)   (independent)

special case is the mean of n independent observations from the same distribution. Since X̄ = (X₁ + X₂ + … + Xₙ) / n, the rules give E(X̄) = μ and Var(X̄) = σ² / n. The standard deviation of the sample mean, σ / √n

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