📚 Binomial Distribution: Properties and Applications | 二项分布的性质与应用
The binomial distribution is one of the most important discrete probability distributions in the IB Mathematics curriculum. It models the number of successes in a fixed number of independent trials, each with the same probability of success. This article explores its definition, properties, and realistic applications.
二项分布是 IB 数学课程中最重要的离散型概率分布之一。它用于刻画在固定次数的独立试验中“成功”发生的次数，其中每次试验的成功概率相同。本文将系统介绍二项分布的定义、性质及实际应用。
1. Definition of the Binomial Distribution | 二项分布的定义
A binomial random variable X is defined as the number of successes in n independent trials, where each trial has only two outcomes: success or failure. If the probability of success on each trial is p, we write X ~ B(n, p).
设随机变量 X 表示在 n 次独立试验中“成功”出现的次数，每次试验只有两种结果：成功或失败。若每次试验成功的概率为 p，则称 X 服从参数为 n 和 p 的二项分布，记作 X ~ B(n, p)。
The probability of obtaining exactly k successes, where k = 0, 1, 2, …, n, is given by the binomial probability formula:
恰好获得 k 次成功（其中 k = 0, 1, 2, …, n）的概率由二项概率公式给出：
P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ
Here C(n, k) is the binomial coefficient, also written ⁿCₖ or ₙCₖ, representing the number of ways to choose k successes from n trials.
其中 C(n, k) 为二项系数，也写作 ⁿCₖ 或 ₙCₖ，表示从 n 次试验中选出 k 次成功的方式数。
2. Conditions for Using the Binomial Model | 二项分布的适用条件
For a situation to be modelled by a binomial distribution, four conditions must hold:
要使用二项分布建模，必须满足以下四个条件：
1. The number of trials n is fixed in advance.
试验次数 n 事先固定。
2. Each trial has exactly two outcomes: success or failure.
每次试验只有两种结果：成功或失败。
3. The probability of success p is the same for every trial.
每次试验成功的概率 p 恒定不变。
4. The trials are independent; the outcome of one trial does not affect another.
各次试验相互独立，一次试验的结果不影响其他试验。
In addition, the binomial variable counts the total number of successes, not the number of trials needed to achieve a success.
此外，二项随机变量统计的是成功总次数，而不是获得一次成功所需的试验次数。
3. The Probability Function and Its Properties | 概率质量函数及其性质
The expression P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ is called the probability mass function of the binomial distribution. It is defined for every integer k from 0 to n.
表达式 P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ 称为二项分布的概率质量函数。它对所有从 0 到 n 的整数 k 都有定义。
Two fundamental properties follow directly from the binomial theorem:
由二项式定理可以直接得到两个基本性质：
∑ᵢ₌₀ⁿ P(X = i) = (p + (1 − p))ⁿ = 1
This shows that the sum of all binomial probabilities equals 1, as expected for any probability distribution.
这说明所有二项概率之和等于 1，这正是任何概率分布都应满足的条件。
4. Mean and Variance | 期望与方差
If X ~ B(n, p), then the expected value and variance have simple closed forms:
若 X ~ B(n, p)，则期望和方差具有简洁的表达式：
E(X) = np
Var(X) = np(1 − p)
The standard deviation is the square root of the variance:
标准差为方差的正平方根：
σ = √(np(1 − p))
These formulas are extremely useful because they allow us to summarise the distribution without listing every probability.
这些公式非常有用，因为它们使我们在不必列出每一个概率的情况下就能概括整个分布的特征。
5. Shape of the Distribution and Mode | 分布形状与最可能值
The shape of a binomial distribution depends on both n and p. When p = 0.5, the distribution is symmetric. When p < 0.5, it is skewed to the right; when p > 0.5, it is skewed to the left.
二项分布的形态取决于 n 与 p。当 p = 0.5 时，分布是对称的；当 p < 0.5 时，分布右偏；当 p > 0.5 时，分布左偏。
As n increases, the distribution becomes more symmetric and more concentrated around its mean np.
随着 n 增大，分布会变得更加对称，并且更集中于均值 np 附近。
The most likely number of successes, called the mode, is approximately np. More precisely, the mode is the integer k that satisfies the inequality (n + 1)p − 1 ≤ k ≤ (n + 1)p.
最可能的成功次数称为众数，其值近似为 np。更精确地，众数是满足不等式 (n + 1)p − 1 ≤ k ≤ (n + 1)p 的整数 k。
6. Cumulative Probabilities | 累积概率
In many problems, we need probabilities of the form P(X ≤ k) or P(X ≥ k). These are known as cumulative probabilities.
许多问题需要计算 P(X ≤ k) 或 P(X ≥ k) 形式的概率，这被称为累积概率。
P(X ≤ k) = ∑ᵢ₌₀ᵏ C(n, i) pⁱ (1 − p)ⁿ⁻ⁱ
For an interval of values, the probability can be written as the difference of two cumulative probabilities:
对于区间概率，可以将其写成两个累积概率之差：
P(a ≤ X ≤ b) = P(X ≤ b) − P(X ≤ a − 1)
Also, the complement rule gives P(X ≥ k) = 1 − P(X ≤ k − 1).
此外，利用补事件可得 P(X ≥ k) = 1 − P(X ≤ k − 1)。
7. Using a Calculator in IB Exams | 在 IB 考试中使用计算器
In IB examinations, a graphical display calculator is essential for binomial calculations. If X ~ B(n, p), use the binomial probability function for exact values and the cumulative function for ranges.
在 IB 考试中，图形计算器对于二项计算至关重要。若 X ~ B(n, p)，使用二项概率函数求精确值，使用累积函数求区间概率。
Typical calculator commands include binompdf(n, p, k) for P(X = k) and binomcdf(n, p, k) for P(X ≤ k).
常见的计算器指令包括：用 binompdf(n, p, k) 求 P(X = k)，用 binomcdf(n, p, k) 求 P(X ≤ k)。
To find P(X ≥ k), use 1 − binomcdf(n, p, k − 1). Always write down the distribution and the probability statement before giving the final answer.
要求 P(X ≥ k) 时，使用 1 − binomcdf(n, p, k − 1)。作答时务必先写出所用分布和概率表达式，再给出最终数值。
8. Real-World Applications | 实际应用
The binomial distribution appears in quality control, medicine, genetics, sports, and many other fields.
二项分布广泛应用于质量控制、医学、遗传学、体育以及许多其他领域。
Example 1: In quality control, a factory inspects 10 items randomly. If each item has a 2% chance of being defective, the number of defective items follows B(10, 0.02). The probability that exactly one item is defective is P(X = 1) = C(10, 1)(0.02)(0.98)⁹.
例 1：在质量控制中，工厂随机检验 10 件产品。若每件产品有 2% 的概率是次品，则次品数量服从 B(10, 0.02)。恰好有一件次品的概率为 P(X = 1) = C(10, 1)(0.02)(0.98)⁹。
Example 2: In genetics, if a plant has a 50% chance of producing a dominant-trait offspring, then among 20 offspring, the number showing the dominant trait follows B(20, 0.5).
例 2：在遗传学中，若某植物产生显性性状后代的概率为 50%，则在 20 个子代中表现显性性状的个数服从 B(20, 0.5)。
Example 3: In medicine, if a vaccine is 90% effective and 50 people are vaccinated, the number of protected people follows B(50, 0.9). This allows researchers to calculate the probability that at least 45 people are protected.
例 3：在医学中，若一种疫苗有效率为 90%，且接种 50 人，则受保护人数服从 B(50, 0.9)。研究者可以据此计算至少 45 人受到保护的概率。
Example 4: In sports, a basketball player has a free-throw success rate of 0.7. Over 12 attempts, the number of made shots follows B(12, 0.7).
例 4：在体育中，某篮球运动员罚球命中率为 0.7。在 12 次罚球中，命中次数服从 B(12, 0.7)。
9. Normal Approximation to the Binomial | 二项分布的正态近似
When n is large, the binomial distribution can be approximated by a normal distribution. A common rule is that the approximation works well when np ≥ 10 and n(1 − p) ≥ 10.
当 n 较大时，二项分布可以用正态分布近似。通常认为，当 np ≥ 10 且 n(1 − p) ≥ 10 时，近似效果较好。
X ≈ N(np, np(1 − p))
Because the binomial distribution is discrete and the normal distribution is continuous, a continuity correction should be applied. For example, P(X ≤ k) is approximated by P(Y ≤ k + 0.5), where Y is the normal variable with the same mean and variance.
由于二项分布是离散分布而正态分布是连续分布，因此需要引入连续性修正。例如，P(X ≤ k) 近似为 P(Y ≤ k + 0.5)，其中 Y 是与原二项分布具有相同均值和方差的正态变量。
10. Common Pitfalls | 常见易错点
Students often make the following mistakes when working with binomial distributions:
学生在处理二项分布时常犯以下错误：
1. Confusing the probability of failure q = 1 − p with the probability of success p.
混淆失败概率 q = 1 − p 与成功概率 p。
2. Forgetting that the trials must be independent for the binomial model to apply.
忽略试验之间必须相互独立这一前提。
3. Using the binomial model when sampling is performed without replacement; the hypergeometric distribution is then more appropriate.
在不放回抽样时使用二项模型；此时超几何分布更为合适。
4. Omitting the complement step when calculating P(X ≥ k).
计算 P(X ≥ k) 时忘记使用补事件。
5. Applying a continuity correction to an exact binomial calculation rather than a normal approximation.
在精确二项计算中错误地加入连续性修正，而连续性修正只应用于正态近似。
11. Exam Tips | 考试技巧
In IB exams, always define the random variable X clearly, state the distribution X ~ B(n, p), and specify the values of n and p
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导