The Supreme Court and Public Policy: Probability, Evidence and Hypothesis Testing | 最高法院与公共政策：概率、证据与假设检验

📚 The Supreme Court and Public Policy: Probability, Evidence and Hypothesis Testing | 最高法院与公共政策：概率、证据与假设检验

Public policy debates often reach the Supreme Court, where decisions on healthcare, immigration, education and civil rights create long-lasting social consequences. This article revises key Edexcel A-Level Mathematics techniques that can model the uncertainty, evidence and risk involved in such judicial and policy decisions.

公共政策辩论常常上诉至最高法院，涉及医疗、移民、教育和公民权利的判决会产生长远的社会影响。本文复习 Edexcel A-Level 数学中可用于模拟此类司法与政策决策的不确定性、证据和风险的关键技术。

1. Probability in the Courtroom: Foundations | 法庭中的概率基础

A probability is a number between 0 and 1 that measures the chance of an event. For example, if a court randomly selects one case from 50 policy appeals, the probability that it concerns education is P(education) = number of education cases / total cases.

概率是介于 0 和 1 之间的数，用来度量事件发生的可能性。例如，如果法院从 50 起政策上诉中随机选取一个案件，则选中教育案件的概率为 P(教育) = 教育案件数 / 总案件数。

Events are mutually exclusive when they cannot happen together. If A is ‘the court upholds a statute’ and B is ‘the court strikes down the same statute’, then P(A ∩ B) = 0 and P(A ∪ B) = P(A) + P(B).

互斥事件是指不能同时发生的事件。若 A 表示“法院维持某法规”，B 表示“法院推翻同一法规”，则 P(A ∩ B)=0，且 P(A ∪ B)=P(A)+P(B)。

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

For independent events, P(A ∩ B) = P(A) × P(B). In legal modelling, two court decisions are often treated as independent only if there is no shared precedent or panel overlap.

对于独立事件，P(A ∩ B)=P(A)×P(B)。在法律建模中，仅当没有共同先例或法官小组重叠时，两次法院判决才可视为独立。

2. Conditional Probability and the ‘Prosecutor’s Fallacy’ | 条件概率与“检察官谬误”

Conditional probability P(A|B) is the probability of A given that B has occurred. The formula is:

条件概率 P(A|B) 是在 B 已发生的条件下 A 发生的概率。公式为：

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

The ‘prosecutor’s fallacy’ confuses P(evidence|innocence) with P(innocence|evidence). For example, if a DNA match has a 1 in 1,000,000 random match probability, that is P(match|innocent). It is not the same as the probability that the defendant is innocent given a match, which depends on the prior probability and other evidence.

“检察官谬误”混淆了 P(证据|无罪) 与 P(无罪|证据)。例如，若 DNA 匹配的随机匹配概率为百万分之一，那是 P(匹配|无罪)。它并不等于给定匹配结果时被告无罪的概率，后者取决于先验概率和其他证据。

In public policy cases, a court may hear statistical evidence such as ‘only 2% of schools fail this standard’. Correct use of conditional probability prevents treating P(fail|school type) as P(school type|fail).

在公共政策案件中，法院可能听取“只有 2% 的学校未达到该标准”的统计证据。正确使用条件概率可以避免将 P(未达标|学校类型) 当作 P(学校类型|未达标)。

3. Bayes’ Theorem: Updating Beliefs from Evidence | 贝叶斯定理：根据证据更新信念

Bayes’ theorem updates a prior probability using new evidence. It is derived directly from the conditional probability formula:

贝叶斯定理利用新证据更新先验概率。它直接由条件概率公式推出：

P(A|B) = [P(B|A) × P(A)] / P(B)

In a judicial review, let A be ‘the policy is discriminatory’ and B be ‘the data show a significant disparity’. Then P(A|B) is the

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