📚 SAT Math Level 2: Probability and Analytic Geometry Key Concepts & Formula Summary | SAT2 数学：概率与解析几何考点及公式总结
This comprehensive guide summarizes all essential probability and analytic geometry formulas you need for the SAT Subject Test Math Level 2. Master counting principles, probability rules, distributions, coordinate geometry, conic sections, and polar coordinates with clear examples.
本指南全面总结了 SAT2 数学（SAT Subject Math Level 2）概率与解析几何所需的全部核心公式与考点，涵盖计数原理、概率法则、分布、坐标几何、圆锥曲线以及极坐标，助你高效备考。
1. Counting Principles | 计数原理
The fundamental counting principle states that if one event can occur in m ways and a second independent event in n ways, then the two events can occur together in m × n ways.
乘法原理：若一个事件有 m 种发生方式，另一个独立事件有 n 种发生方式，则两者共同发生有 m × n 种方式。
A permutation is an arrangement of objects where order matters: P(n, k) = n! / (n – k)!.
排列（顺序有关）：P(n, k) = n! / (n – k)!。
A combination is a selection of objects where order does not matter: C(n, k) = n! / [k!(n – k)!].
组合（顺序无关）：C(n, k) = n! / [k!(n – k)!]。
For repeated items: the number of distinct permutations of n items with n₁ identical of one type, n₂ of another, etc., is n! / (n₁! n₂! …).
有重复物品的排列数：若 n 个物品中有 n₁ 个相同、n₂ 个另一种相同……，不同排列数为 n! / (n₁! n₂! …)。
2. Basic Probability Rules | 基本概率法则
The probability of an event A is P(A) = (number of favorable outcomes) / (total number of outcomes), assuming all outcomes are equally likely.
概率定义：等可能条件下，事件 A 的概率 P(A) = 有利结果数 / 总结果数。
Complement rule: P(A’) = 1 – P(A).
互补事件：P(非A) = 1 – P(A)。
Addition rule for any two events: P(A ∪ B) = P(A) + P(B) – P(A ∩ B). If A and B are mutually exclusive, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).
加法公式：P(A 或 B) = P(A) + P(B) – P(A 且 B)。若 A 与 B 互斥，则 P(A 且 B)=0，即 P(A ∪ B) = P(A) + P(B)。
Multiplication rule for independent events: P(A ∩ B) = P(A) × P(B).
独立事件乘法：P(A 且 B) = P(A) × P(B)。
3. Conditional Probability and Independence | 条件概率与独立性
Conditional probability: P(A|B) = P(A ∩ B) / P(B), provided P(B) ≠ 0.
条件概率：P(A|B) = P(A 且 B) / P(B)，其中 P(B) ≠ 0。
General multiplication rule: P(A ∩ B) = P(B) × P(A|B) = P(A) × P(B|A).
一般乘法公式：P(A 且 B) = P(B) × P(A|B) = P(A) × P(B|A)。
Events A and B are independent if P(A|B) = P(A), or equivalently P(A ∩ B) = P(A)P(B).
独立性检验：若 P(A|B) = P(A)，或 P(A 且 B) = P(A)P(B)，则 A 与 B 独立。
4. Expected Value and Variance | 期望值与方差
For a discrete random variable X with values xᵢ and probabilities pᵢ, the expected value (mean) is μ = E(X) = Σ xᵢ pᵢ.
离散型随机变量 X 的期望值（均值）：μ = E(X) = Σ xᵢ pᵢ。
Variance: Var(X) = Σ (xᵢ – μ)² pᵢ = E(X²) – [E(X)]².
方差：Var(X) = Σ (xᵢ – μ)² pᵢ = E(X²) – [E(X)]²。
Standard deviation: σ = √Var(X).
标准差：σ = √Var(X)。
For any random variables X and Y, E(aX + bY) = aE(X) + bE(Y). If X and Y are independent, Var(aX + bY) = a²Var(X) + b²Var(Y).
期望的线性性质：E(aX + bY) = aE(X) + bE(Y)。若 X 与 Y 独立，Var(aX + bY) = a²Var(X) + b²Var(Y)。
5. Binomial Distribution | 二项分布
A binomial experiment consists of n independent trials, each with two outcomes (success/failure), constant probability of success p. The random variable X = number of successes follows B(n, p).
二项分布：n 次独立试验，每次成功概率 p，X 为成功次数，则 X ~ B(n, p)。
P(X = k) = C(n, k) pᵏ (1-p)ⁿ⁻ᵏ
Mean μ = np, variance σ² = np(1-p).
均值 μ = np，方差 σ² = np(1-p)。
The distribution is symmetric when p = 0.5; it is skewed for other values.
当 p=0.5 时分布对称，否则偏斜。
6. Normal Distribution | 正态分布
The normal distribution is a continuous probability distribution with bell-shaped curve, defined by mean μ and standard deviation σ. The standard normal distribution has μ = 0 and σ = 1.
正态分布是均值为 μ、标准差为 σ 的连续分布，标准正态分布 μ=0、σ=1。
To find probabilities, convert X to a z-score: z = (X – μ) / σ.
标准化：z = (X – μ) / σ。
The empirical rule: approximately 68% of data lie within μ ± σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ.
经验法则：约 68% 的值落在 μ±σ 内，95% 在 μ±2σ 内，99.7% 在 μ±3σ 内。
Use the standard normal table to find the area under the curve for any z-value.
通过标准正态表可查出任意 z 值对应的概率。
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