Pre-U OCR Statistics: Unit Test Mock Paper Analysis | Pre-U OCR 统计：单元测试模拟卷解析

📚 Pre-U OCR Statistics: Unit Test Mock Paper Analysis | Pre-U OCR 统计：单元测试模拟卷解析

This mock unit test covers essential topics in Pre-U OCR Statistics, including probability, discrete and continuous distributions, sampling, and hypothesis testing. Each question is followed by a thorough step-by-step solution in both English and Chinese, helping you refine your understanding and exam technique.

本模拟单元测试涵盖 Pre-U OCR 统计的必考专题：概率、离散与连续分布、抽样及假设检验。每道题后均配有详尽的中英双语解析，逐步推导，助你巩固概念、优化应试策略。

1. Probability and Conditional Probability | 概率与条件概率

A bag contains 5 red balls and 3 green balls. Two balls are drawn at random without replacement. (a) Find the probability that both balls are red. (b) Given that the second ball drawn is red, find the probability that the first ball drawn was also red.

一个袋子里有 5 个红球和 3 个绿球。从中不放回地随机抽取两个球。(a) 求两个都是红球的概率。(b) 已知第二个抽到的是红球，求第一个也是红球的概率。

Part (a) – Use the multiplication rule for dependent events. Let R₁ be the event that the first ball is red, and R₂ that the second is red. P(R₁) = 5/8. After one red is taken, 4 reds remain out of 7 balls, so P(R₂ | R₁) = 4/7. Hence P(R₁ ∩ R₂) = (5/8) × (4/7) = 20/56 = 5/14.

(a) 部分 – 利用相依事件的乘法法则。设 R₁ 为第一次抽到红球，R₂ 为第二次抽到红球。P(R₁) = 5/8。在抽走一个红球后，剩下 7 个球中有 4 个红球，故 P(R₂ | R₁) = 4/7。因此 P(R₁ ∩ R₂) = (5/8) × (4/7) = 20/56 = 5/14。

Part (b) – We need P(R₁ | R₂) = P(R₁ ∩ R₂) / P(R₂). We already have the numerator. To find P(R₂), consider the two ways the second ball can be red: RR or GR (green then red). P(GR) = P(G₁) × P(R₂ | G₁) = (3/8) × (5/7) = 15/56. So P(R₂) = 5/14 + 15/56 = 20/56 + 15/56 = 35/56 = 5/8. Then P(R₁ | R₂) = (5/14) / (5/8) = (5/14) × (8/5) = 40/70 = 4/7.

(b) 部分 – 需要求 P(R₁ | R₂) = P(R₁ ∩ R₂) / P(R₂)。分子已知。为求 P(R₂)，考虑第二次抽到红球的两种情形：先红后红 (RR) 或先绿后红 (GR)。P(GR) = P(G₁) × P(R₂ | G₁) = (3/8) × (5/7) = 15/

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