Modelling the Likelihood of Global Governance | 全球治理可能性的数学建模

📚 Modelling the Likelihood of Global Governance | 全球治理可能性的数学建模

In Edexcel A-Level Mathematics, probability models are often applied to real-world contexts. This article uses the question ‘What is the likelihood of global governance by 2030?’ as a case study to revise conditional probability, Bayes’ theorem, the binomial distribution, normal approximation and hypothesis testing.

在爱德思 A-Level 数学中，概率模型经常应用于现实情境。本文以“2030 年前实现全球治理的可能性有多大？”为案例，复习条件概率、贝叶斯定理、二项分布、正态近似与假设检验。

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1. Defining the Event and Sample Space | 定义事件与样本空间

Let G be the event that a binding global governance framework is achieved by 2030. The sample space S consists of all possible outcomes of international negotiations, so S = {success, failure, partial agreement}.

设 G 表示“到 2030 年达成有约束力的全球治理框架”这一事件。样本空间 S 由国际谈判的所有可能结果组成，因此 S = {成功，失败，部分协议}。

A clear event definition is essential because Edexcel mark schemes require precise use of P(G) and P(G′), where G′ is the complement of G.

清晰的事件定义至关重要，因为爱德思评分标准要求准确使用 P(G) 和 P(G′)，其中 G′ 是 G 的补事件。

P(G) + P(G′) = 1

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2. Estimating Base Rates from Historical Data | 从历史数据估计基准概率

A base rate can be estimated by recording the number of major international treaties that achieved full ratification among all attempts.

基准概率可以通过记录所有尝试中获得全面批准的重大国际条约数量来估计。

If 18 out of 60 comparable climate and trade agreements became fully binding, the relative frequency estimate is P(G) = 18/60 = 0.3.

如果在 60 个可比的气候与贸易协定中有 18 个成为完全有约束力的协定，则相对频率估计为 P(G) = 18/60 = 0.3。

P(G) = 18 ÷ 60 = 0.30

This relative frequency approach links to the Edexcel large data set and sampling ideas.

这种相对频率方法与爱德思大数据集和抽样思想相关。

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3. Conditional Probability and Expert Judgement | 条件概率与专家判断

Experts rarely give a single probability; they give conditional probabilities such as P(G | E), where E is the event that major economies reach a preliminary political consensus.

专家很少给出单一概率；他们给出条件概率，例如 P(G | E)，其中 E 表示主要经济体达成初步政治共识的事件。

The multiplication rule states P(G ∩ E) = P(E) × P(G | E).

乘法法则为 P(G ∩ E) = P(E) × P(G | E)。

P(G | E) = P(G ∩ E) ÷ P(E), P(E) > 0

In Edexcel questions, students often confuse P(G | E) with P(E | G). Drawing a tree diagram helps separate the two directions.

在爱德思考题中，学生经常混淆 P(G | E) 与 P(E | G)。画树状图有助于区分两个方向。

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4. Bayes’ Theorem for Updating Beliefs | 用贝叶斯定理更新信念

When new evidence arrives, Bayes’ theorem updates the prior probability P(G) to a posterior probability.

当新证据出现时，贝叶斯定理将先验概率 P(G) 更新为后验概率。

If E is ‘all major emitters accept a carbon border mechanism’, and we know P(E | G) = 0.8, P(E | G′) = 0.2, with prior P(G) = 0.3, then P(G | E) is:

如果 E 表示“所有主要排放国接受碳边境机制”，已知 P(E | G) = 0.8，P(E | G′) = 0.2，先验 P(G) = 0.3，则 P(G | E) 为：

P(G | E) = [P(E | G) P(G)] ÷ [P(E | G) P(G) + P(E | G′) P(G′)]

Substituting gives P(G | E) = (0.8 × 0.3) ÷ (0.8 × 0.3 + 0.2 × 0.7) = 0.24 ÷ 0.38 ≈ 0.632.

代入得 P(G | E) = (0.8 × 0.3) ÷ (0.8 × 0.3 + 0.2 × 0.7) = 0.24 ÷ 0.38 ≈ 0.632。

This shows that a positive signal can substantially revise the likelihood, a key application of Bayesian reasoning in the Edexcel syllabus.

这表明积极信号可以大幅修正可能性，是爱德思大纲中贝叶斯推理的关键应用。

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5. Independence and Interaction Effects | 独立性与交互效应

Many global governance sub-events, such as security cooperation and trade reform, are not independent. If they were independent, P(A ∩ B) = P(A) × P(B).

许多全球治理子事件（如安全合作与贸易改革）并非独立。如果它们独立，则 P(A ∩ B) = P(A) × P(B)。

Suppose P(security framework) = 0.6 and P(trade reform) = 0.5. Under independence, joint success probability is 0.6 × 0.5 = 0.3.

假设安全框架成功概率为 0.6，贸易改革成功概率为 0.5。在独立条件下，联合成功概率为 0.6 × 0.5 = 0.3。

In reality, positive interaction might make joint success more likely than the product, so an interaction term must be added.

现实中，积极的交互作用可能使联合成功概率大于乘积，因此必须加入

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