Binomial Distribution: Concepts and Applications | 二项分布的概念与应用

📚 Binomial Distribution: Concepts and Applications | 二项分布的概念与应用

The binomial distribution is one of the most important discrete probability distributions in mathematics. 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, conditions, formulas, properties, and real-world applications, with a focus on IB Mathematics expectations.

二项分布是数学中最重要的离散概率分布之一。它用于建模在固定次数的独立试验中成功的次数，每次试验的成功概率相同。本文将深入探讨二项分布的定义、条件、公式、性质及现实应用，紧扣IB数学课程标准。

1. Definition and Conditions | 定义与条件

A random variable \(X\) follows a binomial distribution if it counts the number of successes in \(n\) independent Bernoulli trials, where each trial has success probability \(p\). We write \(X \sim B(n, p)\).

如果随机变量 \(X\) 统计的是 \(n\) 次独立伯努利试验中成功的次数，且每次试验的成功概率为 \(p\)，则称 \(X\) 服从二项分布，记作 \(X \sim B(n, p)\)。

For \(X\) to be binomial, four conditions must hold:

要使 \(X\) 服从二项分布，必须满足以下四个条件：

Fixed number of trials \(n\) — the experiment is repeated a fixed number of times.

试验次数 \(n\) 固定 —— 实验重复固定的次数。
Two possible outcomes per trial — usually called “success” and “failure”.

每次试验只有两种可能结果 —— 通常称为“成功”和“失败”。
Constant probability of success \(p\) — \(p\) is the same for every trial.

成功概率 \(p\) 恒定 —— 每次试验中 \(p\) 保持不变。
Independent trials — 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。

2. Probability Mass Function | 概率质量函数

If \(X \sim B(n, p)\), the probability of getting exactly \(k\) successes is given by:

若 \(X \sim B(n, p)\)，恰好得到 \(k\) 次成功的概率为：

P(X = k) = C(n, k) pᵏ (1 – p)ⁿ⁻ᵏ, k = 0, 1, 2, …, n

Here \(C(n, k)\) is the binomial coefficient, often written as \(\binom{n}{k}\) or \({}_nC_k\), representing the number of ways to choose \(k\) successes from \(n\) trials.

其中 \(C(n, k)\) 是二项式系数，也常写作 \(\binom{n}{k}\) 或 \({}_nC_k\)，表示从 \(n\) 次试验中选出 \(k\) 次成功的方法数。

For example, if a fair coin is tossed 5 times, the probability of getting exactly 3 heads is:

例如，掷一枚均匀硬币5次，恰好出现3次正面的概率为：

P(X = 3) = C(5, 3)(0.5)³(0.5)² = 10 × 0.125 × 0.25 = 0.3125

3. Mean and Variance | 均值与方差

For \(X \sim B(n, p)\), the expected value (mean) and variance are:

对于 \(X \sim B(n, p)\)，其期望（均值）和方差为：

E(X) = np, Var(X) = np(1 – p)

The standard deviation is the square root of the variance: \(\sigma = \sqrt{np(1-p)}\).

标准差是方差的平方根：\(\sigma = \sqrt{np(1-p)}\)。

These formulas are particularly useful in IB examinations, where you may be asked to find the mean and variance directly from \(n\) and \(p\), or to solve for unknown parameters using given moments.

这些公式在IB考试中尤为常用，题目可能要求直接从 \(n\) 和 \(p\) 求均值与方差，或利用给定的矩来求解未知参数。

4. Mode and Shape | 众数与形状

The mode of a binomial distribution is the value of \(k\) that maximises \(P(X = k)\). It can be found using:

二项分布的众数是使 \(P(X = k)\) 达到最大的 \(k\) 值，可通过以下方式确定：

Mode = ⌊(n + 1)p⌋ (if (n+1)p is not an integer)

If \((n+1)p\) is an integer, then both \((n+1)p\) and \((n+1)p – 1\) are modes.

如果 \((n+1)p\) 是整数，则 \((n+1)p\) 和 \((n+1)p – 1\) 都是众数。

The distribution is symmetric when \(p = 0.5\). When \(p < 0.5\), the distribution is skewed to the right (positive skew); when \(p > 0.5\), it is skewed to the left (negative skew). As \(n\) increases, the distribution becomes more symmetric and bell-shaped.

当 \(p = 0.5\) 时，分布对称；当 \(p < 0.5\) 时，分布右偏（正偏）；当 \(p > 0.5\) 时，分布左偏（负偏）。随着 \(n\) 增大，分布越来越对称，呈钟形。

5. Cumulative Probabilities | 累积概率

In IB problems, you often need \(P(X \le k)\) or \(P(X \ge k)\) rather than individual probabilities. For example:

在IB题目中，经常需要计算 \(P(X \le k)\) 或 \(P(X \ge k)\)，而非仅仅单个值的概率。例如：

If \(X \sim B(10, 0.3)\), find \(P(X \le 4)\). This can be computed as the sum \(P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)\).

若 \(X \sim B(10, 0.3)\)，求 \(P(X \le 4)\)。这可以写成 \(P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)\) 的和。

On a GDC (graphical display calculator), this is usually obtained using the built-in binomial cumulative distribution function (e.g. binomCdf).

在图形计算器（GDC）上，通常使用内置的二项累积分布函数（例如 binomCdf）直接得到结果。

Remember the complement rule: \(P(X \ge k) = 1 – P(X \le k-1)\). This is especially useful when \(k\) is large.

注意补集法则：\(P(X \ge k) = 1 – P(X \le k-1)\)。当 \(k\) 较大时这尤其方便。

6. Using the Calculator | 使用计算器

IB students are expected to use a GDC or a scientific calculator with statistical functions. For \(X \sim B(n, p)\):

IB学生需要会使用带有统计功能的图形计算器或科学计算器。对于 \(X \sim B(n, p)\)：

To find \(P(X = k)\), use the binomial probability command (e.g. binompdf or binomPdf).

求 \(P(X = k)\)，使用二项概率命令（例如 binompdf 或 binomPdf）。
To find \(P(X \le k)\), use the binomial cumulative command (e.g. binomcdf).

求 \(P(X \le k)\)，使用二项累积命令（例如 binomcdf）。
For \(P(X \ge k)\), compute \(1 – \text{binomcdf}(n, p, k-1)\).

求 \(P(X \ge k)\)，计算 \(1 – \text{binomcdf}(n, p, k-1)\)。

Always check the syntax of your calculator model, as command names may vary.

务必检查所用计算器型号的语法，因为命令名称可能有所不同。

7. Applying the Binomial Model | 应用二项模型

A common IB exam question describes a real-world scenario and asks whether a binomial distribution is appropriate. The following checklist helps:

IB考试中常见的一类题目是描述一个现实情境，并要求判断二项分布是否适用。以下检查清单很有帮助：

Condition	Question to ask	条件	应问的问题
Fixed n	Is the number of trials predetermined?	次数固定	试验次数是否预先确定？
Binary outcomes	Can each trial be classified as success/failure?	结果二元	每次试验是否能分为成功/失败？
Constant p	Does the success probability stay the same?	概率恒定	成功概率是否保持不变？
Independence	Are the trials independent of each other?	相互独立	各次试验是否相互独立？

If sampling is done without replacement from a finite population, the independence condition fails. In some cases, a hypergeometric distribution is more appropriate.

如果从有限总体中不放回抽样，独立性条件不成立。此时超几何分布可能更合适。

8. Real-World Examples | 现实应用举例

Binomial distributions appear in many fields:

二项分布广泛应用于许多领域：

Quality control: In a batch of 100 products, the number of defective items, given a defect rate of 2%, is \(B(100, 0.02)\).

质量控制： 一批100件产品中，若次品率为2%，则次品数量服从 \(B(100, 0.02)\)。
Medicine: The number of patients who recover after a treatment, out of 50 patients, with recovery probability 0.9, is \(B(50, 0.9)\).

医学： 50名患者接受治疗后康复的人数，若康复概率为0.9，则服从 \(B(50, 0.9)\)。
Sports: The number of free throws made out of 10 attempts, if the player’s free-throw success rate is 0.7, is \(B(10, 0.7)\).

体育： 一名罚球命中率为0.7的球员在10次罚球中命中的次数服从 \(B(10, 0.7)\)。
Surveys: In a random poll of 200 voters, the number supporting a candidate, if 55% support the candidate, is \(B(200, 0.55)\).

调查： 在200名选民的随机民意调查中，若55%支持某候选人，则支持人数服从 \(B(200, 0.55)\)。

9. Normal Approximation to the Binomial | 二项分布的正态近似

When \(n\) is large and \(p\) is not too close to 0 or 1, the binomial distribution can be approximated by a normal distribution:

当 \(n\) 较大且 \(p\) 不太接近0或1时，二项分布可以用正态分布近似：

X \sim B(n, p) \approx N(np, np(1-p))

A common rule is that the normal approximation is reasonable if \(np \ge 5\) and \(n(1-p) \ge 5\).

通常的判断标准是：若 \(np \ge 5\) 且 \(n(1-p) \ge 5\)，则正态近似效果较好。

When using a normal approximation, a continuity correction is applied. For example, \(P(X \le k)\) is approximated by \(P(Y \le k + 0.5)\), where \(Y \sim N(np, np(1-p))\).

使用正态近似时需要进行连续性修正。例如，\(P(X \le k)\) 用 \(P(Y \le k + 0.5)\) 来近似，其中 \(Y \sim N(np, np(1-p))\)。

This topic is particularly relevant in the IB AI and AA courses, as it connects discrete and continuous probability models.

该主题在IB的AI和AA课程中尤为重要，因为它将离散概率模型与连续概率模型联系起来。

10. Distinguishing Binomial from Other Distributions | 区分二项分布与其他分布

In IB exams, you must choose the correct distribution for a scenario. Here are key differences:

在IB考试中，你必须根据情境选择正确的分布。以下是一些关键区别：

Distribution	Counts	Conditions	分布	统计对象	条件
Binomial	Number of successes in a fixed n	Independent trials, constant p	二项分布	固定次数中的成功次数	独立试验，概率恒定
Geometric	Trial number of the first success	No fixed n; constant p	几何分布	首次成功所在试验次数	无固定次数；概率恒定
Poisson	Number of events in an interval	Independent, rare events; rate λ	泊松分布	区间内事件次数	独立、稀有事件；速率 λ

For example, if you toss a coin until you get a head, the number of tosses follows a geometric distribution, not binomial.

例如，一直掷硬币直到出现正面，掷的次数服从几何分布，而不是二项分布。

11. Hypothesis Testing with Binomial | 二项分布的假设检验

In IB Mathematics Analysis and Approaches (AA) and Applications and Interpretation (AI), you may be asked to perform a hypothesis test using a binomial model. The null hypothesis typically states \(H_0: p = p_0\), and the alternative is one-tailed or two-tailed.

在IB数学分析与方法（AA）以及应用与解释（AI）中，你可能会被要求使用二项模型进行假设检验。原假设通常为 \(H_0: p = p_0\)，备择假设为单尾或双尾。

For a one-tailed test \(H_1: p < p_0\), you calculate \(P(X \le x)\) given \(X \sim B(n, p_0)\). If this probability is less than the significance level (e.g. 5%), you reject \(H_0\).

对于单尾检验 \(H_1: p < p_0\)，在 \(X \sim B(n, p_0)\) 下计算 \(P(X \le x)\)。如果该概率小于显著性水平（如5%），则拒绝 \(H_0\)。

Similarly, for \(H_1: p > p_0\), use \(P(X \ge x)\). For two-tailed tests, double the smaller tail probability and compare with the significance level.

类似地，对于 \(H_1: p > p_0\)，使用 \(P(X \ge x)\)。对于双尾检验，将较小的尾部概率加倍并与显著性水平比较。

Make sure to state the hypotheses in terms of the population proportion \(p\), not the sample proportion.

务必用总体比例 \(p\) 来表述假设，而不是样本比例。

12. Common Pitfalls and Tips | 常见错误与技巧

Students often make these mistakes when working with binomial distributions:

学生在处理二项分布时常犯以下错误：

Forgetting to check the conditions before applying the binomial model.

在应用二项模型前忘记检查条件。
Confusing \(P(X \ge k)\) with \(1 – P(X \le k)\). Remember the correct complement: \(P(X \ge k) = 1 – P(X \le k-1)\).

混淆 \(P(X \ge k)\) 与 \(1 – P(X \le k)\)。正确补集为 \(P(X \ge k) = 1 – P(X \le k-1)\)。
Using \(p\) instead of \(1-p\) when counting failures as successes.

把失败当作成功计算时，误用 \(p\) 而不用 \(1-p\)。
Forgetting that in a two-tailed binomial test, the \(p\)-value is twice the smaller tail probability.

在双尾二项检验中，忘记 \(p\)-值是较小尾部概率的两倍。

Tip: Always write down \(X \sim B(n, p)\) first, define what success means, and identify \(n\) and \(p\) before doing any calculations.

技巧：先写出 \(X \sim B(n, p)\)，明确“成功”的含义，并在计算前确定 \(n\) 和 \(p\)。

Published by TutorHao | IB Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Binomial Distribution: Concepts and Applications | 二项分布的概念与应用