📚 Binomial Distribution: Model and Application Essentials | 二项分布的模型与应用要点
The binomial distribution is one of the most widely used discrete probability models in statistics. It describes the number of successes in a fixed number of independent trials, each with the same probability of success, and forms the basis for many real-world decisions and examinations.
二项分布是统计学中最常用的离散概率模型之一。它描述在固定次数的独立试验中，每次试验成功概率相同的情况下“成功”次数的分布，也是许多实际问题判断和考试的重要基础。
1. What Is a Binomial Distribution? | 什么是二项分布
A binomial distribution is the probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes/no question. If the random variable X counts the number of successes, we write X ~ B(n, p), where n is the number of trials and p is the probability of success on a single trial.
二项分布是 n 次独立试验中“成功”次数的概率分布，每次试验都是一个“是/否”问题。若随机变量 X 表示成功次数，则记作 X ~ B(n, p)，其中 n 是试验次数，p 是单次试验成功的概率。
The possible values of X are the integers from 0 to n, so the binomial distribution is a discrete distribution with a finite sample space.
X 的可能取值为从 0 到 n 的整数，因此二项分布是取值有限的离散分布。
2. Conditions for a Binomial Model | 二项模型的条件
For a situation to be modelled by a binomial distribution, four conditions must be satisfied simultaneously. These conditions are often tested in examination questions, so it is essential to check them carefully.
一个情况要能用二项分布建模，必须同时满足四个条件。考试中经常考查这些条件，因此务必仔细验证。
There is a fixed number of trials, n.
试验次数 n 固定不变。
Each trial has exactly two possible outcomes: success or failure.
每次试验只有两种可能结果：“成功”或“失败”。
The trials are independent of one another.
各次试验之间相互独立。
The probability of success p is the same for every trial.
每次试验成功的概率 p 保持不变。
3. Probability Mass Function | 概率质量函数
If X ~ B(n, p), the probability of observing exactly r successes is given by the binomial probability mass function. Here C(n, r) is the binomial coefficient, also written as n choose r.
若 X ~ B(n, p)，则恰好出现 r 次成功的概率由二项概率质量函数给出。其中 C(n, r) 是组合数，也记为 n 选 r。
P(X = r) = C(n, r) pr (1 - p)n - r
for r = 0, 1, 2, ..., n. The value C(n, r) = n! / [r! (n - r)!] counts the number of ways to choose r successes from n trials.
其中 r = 0, 1, 2, ... , n。组合数 C(n, r) = n! / [r! (n - r)!] 表示从 n 次试验中选出 r 次成功的方式数。
Example: For X ~ P(X = 2). Here C(5, 2) = 10, so P(X = 2) = 10 × 0.4² × 0.6³ = 10 × 0.16 × 0.216 = 0.3456.
示例：设 X ~ B(5, 0.4)，求 P(X = 2)。这里 C(5, 2) = 10，因此 P(X = 2) = 10 × 0.4² × 0.6³ = 10 × 0.16 × 0.216 = 0.3456。
4. Mean, Variance and Standard Deviation | 均值、方差与标准差
The binomial distribution has simple formulas for its mean, variance and standard deviation. These formulas are derived from the sum of n independent Bernoulli random variables, each with mean p and variance p(1 - p).
二项分布有简洁的均值、方差和标准差公式。这些公式来源于 n 个独立伯努利随机变量之和，每个伯努利变量的均值为 p，方差为 p(1 - p)。
Mean: μ = np
Variance: σ² = np(1 - p)
Standard deviation: σ = √[np(1 - p)]
tremely useful in both descriptive statistics and statistical inference. For example, if X ~ B(200, 0.3), then μ = 60 and σ = √200 × 0.3 × 0.7 = √42 ≈ 6.48.
这些公式在描述统计和统计推断中非常有用。例如，若 X ~ B(200, 0.3)，则 μ = 60，σ = √200 × 0.3 × 0.7 = √42 ≈ 6.48。
5. Shape and Skewness | 分布形态与偏态
ion is perfectly symmetric around its mean. When p < 0.5, the distribution is skewed to the right, with a longer tail on the higher-value side. When p > 0.5, it is skewed to the left.
二项分布的形态主要取决于成功概率 p。当 p = 0.5 时，分布关于均值完全对称。当 p < 0.5 时，分布右偏，右侧尾部较长；当 p > 0.5 时，分布左偏。
As n increases, the distribution becomes more symmetric and smoother, approaching a normal-shaped curve even if p is not equal to 0.5. This behaviour is the foundation of the normal approximation discussed below.
随着 n 增大，分布会变得更对称、更平滑，即使 p 不等于 0.5，也逐渐接近正态曲线。这一性质是下文正态近似的基础。
6. Cumulative Probabilities and Tables | 累积概率与查表
ility. The cumulative distribution function is defined as the sum of point probabilities from 0 up to r.
在许多问题中，我们需要 P(X ≤ r) 或 P(X ≥ r)，而不只是某一点的概率。累积分布函数定义为从 0 到 r 的点概率之和。
P(X ≤ r) = Σk=0r C(n, k) pk (1 - p)n - k
or or software can compute these sums directly. Remember that P(X ≥ r) = 1 - P(X ≤ r - 1).
统计表通常会给出常见 n 和 p 下的 P(X ≤ r)。计算器或软件可以直接计算这些求和。注意 P(X ≥ r) = 1 - P(X ≤ r - 1)。
7. Applications in Real Life | 实际应用
The binomial model appears in many fields, including quality control, medicine, finance, sports and education. Whenever a process yields a binary outcome and every observation is independent with constant probability, binomial calculations are appropriate.
二项模型广泛应用于质量控制、医学、金融、体育和教育等领域。只要一个过程产生二值结果、每次观测独立且概率恒定，就可以用二项分布进行计算。
Quality control: counting defective items in a random sample of n products, where each item is defective or not.
质量控制：在 n 件随机抽取的产品中统计次品数，每件产品要么是次品，要么不是次品。
Medicine: the number of patients who recover after receiving a treatment, assuming each patient's outcome is independent.
医学：接受治疗后痊愈的患者人数，假设每位患者的治疗结果是独立的。
Sports: the number of free throws scored by a basketball player out of n attempts with a constant free-throw percentage.
体育：篮球运动员在 n 次罚球中命中的次数，假设罚球命中率恒定。
8. Normal Approximation to the Binomial | 正态近似
mial distribution B(n, p) can be approximated by a normal distribution with the same mean and variance.
当 n 很大时，直接计算二项概率非常繁琐。若 np ≥ 5 且 n(1 - p) ≥ 5，则二项分布 B(n, p) 可用具有相同均值和方差的正态分布来近似。
X ~ B(n, p) ≈ N(np, np(1 - p))
is required. For example, P(X ≤ r) becomes P(Y ≤ r + 0.5) where Y is the normal variable.
由于二项分布是离散的而正态分布是连续的，因此需要进行连续性修正。例如，P(X ≤ r) 写成 P(Y ≤ r + 0.5)，其中 Y 是正态变量。
Example: X ~ B(120, 0.5). Then μ = 60, σ = √30 ≈ 5.477. To approximate P(X ≤ 55), we compute P(Y ≤ 55.5) using Y ~ N(60, 30).
示例：X ~ B(120, 0.5)，则 μ = 60，σ = √30 ≈ 5.477。要近似 P(X ≤ 55)，用 Y ~ N(60, 30) 计算 P(Y ≤ 55.5)。
9. Poisson Approximation to the Binomial | 泊松近似
A common rule of thumb is n ≥ 50 and p ≤ 0.1, or more conservatively n ≥ 100 and np ≤ 10.
当 n 很大而 p 很小时，二项概率可以用参数 λ = np 的泊松分布来近似。常用规则是 n ≥ 50 且 p ≤ 0.1，或更保守地 n ≥ 100 且 np ≤ 10。
P(X = r) ≈ e-λ λr / r!
This approximation is especially useful in rare-event modelling, such as the number of accidents in a period or the number of defective items in a large batch.
该近似特别适用于稀有事件建模，例如一段时间内事故发生的次数，或大批量产品中的次品数量。
10. Hypothesis Testing with the Binomial Distribution | 二项分布的假设检验
p₀ and an alternative hypothesis H₁, then calculate the probability of obtaining a result as extreme as the observed value under H₀.
二项分布常用于对比例进行假设检验。我们建立原假设 H₀: p = p₀ 和备择假设 H₁，然后计算在原假设成立时，观测到与当前结果一样极端的结果的概率。
ne-tailed test uses P(X ≥ c) or P(X ≤ c), while a two-tailed test considers both extremes.
临界区域的选择要使第一类错误的总概率至多为显著性水平 α。单尾检验使用 P(X ≥ c) 或 P(X ≤ c)，双尾检验则同时考虑两端极值。
Example: times and shows heads 14 times. To test whether the coin is biased towards heads at the 5% significance level, let X ~ B(20, 0.5) under H₀. We compute P(X ≥ 14) = 0.0577 > 0.05, so we do not reject H₀.
示例：将一枚硬币抛 20 次，出现正面 14 次。在 5% 显著性水平下检验硬币是否偏向正面，设 H₀ 下 X ~ B(20, 0.5)，计算 P(X ≥ 14) = 0.0577 > 0.05，因此不拒绝原假设。
11. Common Mistakes and Tips | 常见错误与提示
Students often lose marks in binomial problems because of small but critical misunderstandings. The following tips help you avoid the most common pitfalls.
学生在二项分布问题中常因一些细小但关键的误解而失分。以下提示帮助你避免最常见的陷阱。
Check the four conditions carefully before applying the binomial model.
在套用二项模型前，仔细检查四个条件是否全部满足。
Do not confuse p with q = 1 - p. Always identify which outcome is being counted as “success”.
不要把 p 和 q = 1 - p 混淆。始终明确哪个结果被记为“成功”。
When using a normal approximation, always apply a continuity correction.
使用正态近似时，务必进行连续性修正。
In hypothesis testing, state the distribution, the null and alternative hypotheses, and the conclusion in context.
在假设检验中，写明分布、原假设和备择假设，并结合实际问题给出结论。
12. Conclusion | 总结
The binomial distribution is a powerful and versatile model for counting successes in independent trials. Its simple formulas for mean and variance, together with well-established approximation methods, make it an essential tool for both coursework and real-world applications.
二项分布是统计独立试验中成功次数的强大且灵活的模型。其简洁的均值和方差公式，加上成熟的正态近似和泊松近似方法，使它成为课程学习和实际应用中的重要工具。
Mastering the conditions, probability function, cumulative probabilities, and approximation techniques will help you solve a wide range of problems confidently.
掌握二项分布的条件、概率函数、累积概率和近似技术，将帮助你自信地解决各种相关问题。
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