IB Mathematics: Conditions and Typical Applications of Binomial Distribution | IB数学：二项分布的条件与典型应用

📚 IB Mathematics: Conditions and Typical Applications of Binomial Distribution | IB数学：二项分布的条件与典型应用

The binomial distribution is one of the most important discrete probability distributions in IB Mathematics. It models the number of successes in a fixed number of independent trials, each with the same probability of success. Understanding its conditions and typical applications is essential for solving exam problems and interpreting real-world data.

二项分布是IB数学中最重要的离散概率分布之一。它用于模拟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。理解其条件与典型应用，对于解答考试题目和解释现实数据至关重要。

1. What Is the Binomial Distribution? | 什么是二项分布？

The binomial distribution is a discrete probability distribution that describes the number of successes in a sequence of \(n\) independent experiments, each with two possible outcomes. In IB notation, we write \(X \sim B(n, p)\), where \(n\) is the number of trials and \(p\) is the probability of success on a single trial.

二项分布是一种离散概率分布，用于描述在一系列 \(n\) 次独立试验中成功的次数，每次试验只有两种可能结果。在IB记法中，我们写作 \(X \sim B(n, p)\)，其中 \(n\) 是试验次数，\(p\) 是单次试验成功的概率。

For example, tossing a coin 10 times and counting the number of heads can be modelled by a binomial distribution if each toss is independent and the probability of heads remains 0.5.

例如，抛一枚硬币10次并统计正面朝上的次数，如果每次抛掷相互独立且正面概率保持0.5，则可以用二项分布来建模。

2. Conditions for a Binomial Distribution | 二项分布的条件

Before applying the binomial model, you must verify that four conditions hold. These conditions are frequently tested in IB papers, especially in Paper 2 and the interpretation questions.

在应用二项模型之前，必须验证四个条件是否成立。这些条件在IB试卷中经常考查，尤其是在Paper 2和解释性问题中。

Fixed number of trials: The experiment is repeated a fixed number of times, denoted by \(n\).

固定试验次数： 试验重复固定的次数，记为 \(n\)。
Two possible outcomes: Each trial has only two outcomes, usually called success and failure.

只有两种可能结果： 每次试验只有两种结果，通常称为成功与失败。
Independent trials: The outcome of one trial does not affect the outcome of another.

各次试验独立： 一次试验的结果不影响另一次试验的结果。
Constant probability: The probability of success \(p\) is the same for every trial.

概率恒定： 每次试验成功的概率 \(p\) 都相同。

If any of these conditions is not met, the binomial distribution is not appropriate. For example, sampling without replacement from a finite population changes the probability, so the hypergeometric distribution may be more suitable.

如果这些条件中任何一条不满足，二项分布就不适用。例如，从有限总体中不放回抽样会改变概率，此时超几何分布可能更合适。

3. Probability Mass Function and Parameters | 概率质量函数与参数

For a binomial random variable \(X \sim B(n, p)\), the probability of getting exactly \(r\) successes is given by the formula:

对于二项随机变量 \(X \sim B(n, p)\)，恰好得到 \(r\) 次成功的概率由以下公式给出：

P(X = r) = C(n, r) p^r (1 – p)^{n – r}, r = 0, 1, 2, …, n

Here \(C(n, r)\) is the binomial coefficient, also written as ⁿCᵣ or \(n\) choose \(r\), which counts the number of ways to choose \(r\) successes among \(n\) trials.

其中 \(C(n, r)\) 是二项式系数，也可写作 ⁿCᵣ 或 \(n\) choose \(r\)，它表示在 \(n\) 次试验中选出 \(r\) 次成功的方式数。

The parameter \(n\) is a positive integer, and \(p\) must satisfy \(0 \le p \le 1\). The distribution is discrete, so its probabilities sum to 1 over all possible values of \(r\).

参数 \(n\) 是正整数，\(p\) 必须满足 \(0 \le p \le 1\)。该分布是离散的，因此所有可能值 \(r\) 的概率之和为1。

4. Mean and Variance | 期望与方差

For a binomial distribution \(X \sim B(n, p)\), the expected value (mean) and variance have simple formulas that appear frequently in IB exams.

对于二项分布 \(X \sim B(n, p)\)，期望（均值）和方差具有简单公式，这些在IB考试中经常出现。

E(X) = np

Var(X) = np(1 – p)

The standard deviation is the square root of the variance: \(\sigma = \sqrt{np(1-p)}\). These formulas are useful for solving problems about expected number of successes and for checking whether a given distribution is plausible.

标准差是方差的正平方根：\(\sigma = \sqrt{np(1-p)}\)。这些公式可用于求解期望成功次数，并检验给定分布是否合理。

For example, if a multiple-choice test has 20 questions and each has 4 options, the expected number of correct answers by pure guessing is \(np = 20 \times 0.25 = 5\).

例如，若一次测验有20道选择题，每题4个选项，纯猜测期望答对的题目数为 \(np = 20 \times 0.25 = 5\)。

5. Shape of the Binomial Distribution | 二项分布的图形形状

The shape of a binomial distribution depends on \(n\) and \(p\). When \(p = 0.5\), the distribution is symmetric. When \(p < 0.5\), it is skewed to the right (positive skew); when \(p > 0.5\), it is skewed to the left.

二项分布的图形取决于 \(n\) 和 \(p\)。当 \(p = 0.5\) 时，分布对称；当 \(p < 0.5\) 时，右偏（正偏）；当 \(p > 0.5\) 时，左偏。

As \(n\) increases, the distribution becomes more symmetric and approaches a normal curve, provided \(p\) is not too close to 0 or 1. This is the basis for the normal approximation to the binomial distribution.

随着 \(n\) 增大，分布变得更对称，并逐渐接近正态曲线，只要 \(p\) 不太接近0或1。这是二项分布正态近似的基础。

For small \(n\), the distribution may be quite irregular. IB questions often ask you to describe the shape or compare probabilities using the probability mass function.

当 \(n\) 较小时，分布可能很不规则。IB题目常要求描述形状或利用概率质量函数比较概率大小。

6. Typical Applications in Real Life | 现实生活中的典型应用

The binomial distribution appears in many real-world situations. Here are some classic examples that are also common in IB exam questions.

二项分布在许多现实情境中都会出现。以下是一些经典示例，也是IB考题中常见的背景。

Quality control: A factory produces items, and each item is defective with probability \(p\). If we inspect \(n\) randomly chosen items, the number of defective items follows a binomial distribution, assuming the production process is stable and items are independent.

质量控制： 工厂生产产品，每件产品有概率 \(p\) 为次品。若随机抽取 \(n\) 件进行检验，在工艺稳定且产品相互独立的假设下，次品件数服从二项分布。

Medicine and biology: In a clinical trial, each patient has probability \(p\) of responding to a treatment. The number of responders among \(n\) patients is binomial if patients are independent.

医学与生物学： 在临床试验中，每位患者对治疗有概率 \(p\) 产生反应。若患者相互独立，则 \(n\) 名患者中产生反应的人数服从二项分布。

Sports: A basketball player has a free-throw success rate of 80%. In 10 free throws, the number of successful shots can be modelled by \(B(10, 0.8)\), if each shot is independent.

体育： 一名篮球运动员罚球命中率为80%。在10次罚球中，命中次数可用 \(B(10, 0.8)\) 建模，前提是每次投篮相互独立。

Surveys and polls: If 60% of voters support a policy, the number of supporters in a random sample of 50 voters can be modelled binomially, provided sampling is random and the population is large enough.

调查与民意测验： 如果60%的选民支持某项政策，在随机抽取的50名选民中支持者人数可用二项分布建模，前提是抽样随机且总体足够大。

7. Binomial vs Bernoulli Distribution | 二项分布与伯努利分布

A Bernoulli distribution is a special case of the binomial distribution with \(n = 1\). It has only one trial, so the random variable takes the value 1 with probability \(p\) and 0 with probability \(1 – p\).

伯努利分布是二项分布的特例，此时 \(n = 1\)。它只有一次试验，因此随机变量以概率 \(p\) 取值1，以概率 \(1 – p\) 取值0。

For a Bernoulli random variable, \(E(X) = p\) and \(Var(X) = p(1-p)\). In IB, you should recognise that a binomial distribution is the sum of \(n\) independent and identically distributed Bernoulli variables.

对于伯努利随机变量，\(E(X) = p\)，\(Var(X) = p(1-p)\)。在IB中，你应该认识到二项分布是 \(n\) 个独立同分布伯努利变量之和。

This relationship helps explain the formulas for the mean and variance: the sum of independent variables has mean equal to the sum of means, and variance equal to the sum of variances.

这一关系有助于解释均值和方差公式：独立变量之和的均值等于各均值之和，方差等于各方差之和。

8. Normal Approximation to 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 \(B(n, p)\) can be approximated by a normal distribution with mean \(np\) and variance \(np(1-p)\). A common rule is that both \(np \ge 5\) and \(n(1-p) \ge 5\).

当 \(n\) 很大且 \(p\) 不太接近0或1时，二项分布 \(B(n, p)\) 可用正态分布近似，其均值为 \(np\)，方差为 \(np(1-p)\)。常用规则是要求 \(np \ge 5\) 且 \(n(1-p) \ge 5\)。

In IB applications, this approximation is used when calculating probabilities for ranges of values, such as \(P(X \le k)\) or \(P(a \le X \le b)\). A continuity correction is often applied because the binomial is discrete while the normal is continuous.

在IB应用中，此近似用于计算区间概率，例如 \(P(X \le k)\) 或 \(P(a \le X \le b)\)。由于二项分布是离散的而正态分布是连续的，通常需要进行连续性修正。

For example, \(X \sim B(100, 0.5)\). Then \(E(X) = 50\), \(Var(X) = 25\), so \(X\) is approximately \(N(50, 25)\). To find \(P(X \le 45)\), use \(P(X \le 45.5)\) in the normal approximation.

例如，\(X \sim B(100, 0.5)\)，则 \(E(X) = 50\)，\(Var(X) = 25\)，因此 \(X\) 近似 \(N(50, 25)\)。要求 \(P(X \le 45)\)，在正态近似中使用 \(P(X \le 45.5)\)。

9. Common Mistakes and How to Avoid Them | 常见错误与规避方法

Many students lose marks in IB exams by applying the binomial distribution without checking its conditions, or by confusing the probability \(P(X = r)\) with the cumulative probability \(P(X \le r)\).

许多学生在IB考试中因未检查条件而直接套用二项分布，或混淆 \(P(X = r)\) 与累积概率 \(P(X \le r)\)，从而丢分。

Mistake 1: Using the binomial model when sampling without replacement from a small population. This changes the probability after each draw.

错误1： 在从较小总体中不放回抽样时使用二项分布。每次抽取后概率会改变。

Mistake 2: Forgetting to define the success clearly. For example, if \(p\) is the probability of failure, the binomial formula must use the probability of success consistently.

错误2： 忘记明确定义“成功”。例如，若 \(p\) 是失败概率，则二项公式中必须一致地使用成功概率。

Mistake 3: Using \(P(X = r)\) for “at least” questions. You must use \(P(X \ge r) = 1 – P(X \le r-1)\).

错误3： 在“至少”型问题中使用 \(P(X = r)\)。应使用 \(P(X \ge r) = 1 – P(X \le r-1)\)。

To avoid these errors, always write down the distribution clearly, check the four conditions, and identify whether the question asks for an exact value or a cumulative range.

为避免这些错误，应明确写出分布，检查四个条件，并判断题目要求的是精确值还是累积区间。

10. Using GDC and Statistical Tables | 使用图形计算器与统计表

In IB Mathematics, you are allowed to use a GDC (graphical display calculator) for binomial probabilities. The calculator functions usually require the values of \(n\), \(p\), and \(r\). You can find both \(P(X = r)\) and \(P(X \le r)\) directly.

在IB数学考试中，允许使用图形计算器（GDC）计算二项概率。计算器函数通常需要输入 \(n\)、\(p\) 和 \(r\) 的值。你可以直接求出 \(P(X = r)\) 和 \(P(X \le r)\)。

For example, on a typical GDC, the binomial probability distribution function is often denoted by binompdf(n, p, r), and the cumulative distribution function by binomcdf(n, p, r).

例如，在常见的GDC上，二项概率分布函数通常记为 binompdf(n, p, r)，累积分布函数记为 binomcdf(n, p, r)。

When using the calculator, be careful with the input order and the meaning of the output. Some calculators give \(P(X \le r)\), while others give \(P(X < r)\). Always double-check the syntax and the result.

使用计算器时，注意输入顺序和输出的含义。有些计算器给出 \(P(X \le r)\)，有些给出 \(P(X < r)\)。务必检查语法和结果。

11. Worked Example | 典型例题精讲

Let us solve a typical IB-style problem. A biased coin has probability 0.6 of landing on heads. The coin is tossed 8 times. Let \(X\) be the number of heads.

让我们解答一道典型的IB风格题。一枚不均匀硬币正面朝上的概率为0.6。将这枚硬币抛掷8次。设 \(X\) 为正面朝上的次数。

First, identify the distribution: \(X \sim B(8, 0.6)\). This is valid because each toss is independent, there are two outcomes, the number of trials is fixed at 8, and the success probability is constant.

首先，确定分布：\(X \sim B(8, 0.6)\)。这是合理的，因为每次抛掷独立，只有两种结果，试验次数固定为8，成功概率恒定。

(a) Find \(P(X = 5)\). Using the formula:

(a) 求 \(P(X = 5)\)。使用公式：

P(X = 5) = C(8, 5) × 0.6⁵ × 0.4³ = 56 × 0.07776 × 0.064 ≈ 0.2787

(b) Find \(P(X \ge 6)\). This equals \(P(X = 6) + P(X = 7) + P(X = 8)\). Using the GDC or manual calculation:

(b) 求 \(P(X \ge 6)\)。它等于 \(P(X = 6) + P(X = 7) + P(X = 8)\)。使用GDC或手算：

P(X \ge 6) = 0.2090 + 0.0896 + 0.0168 ≈ 0.3154

This worked example shows how to apply the binomial formula step by step and how to combine probabilities for ranges of values.

这个例题展示了如何逐步应用二项公式，以及如何组合区间概率。

12. Summary | 总结

The binomial distribution is a powerful tool for modelling counts of successes in fixed, independent trials with constant probability. The four conditions — fixed number of trials, two outcomes, independence, and constant \(p\) — are essential requirements.

二项分布是对固定、独立且概率恒定试验中成功次数进行建模的有力工具。四个条件——固定试验次数、两种结果、独立性和恒定概率 \(p\)——是必要前提。

You should know the probability mass function, the mean and variance formulas, and be able to apply them to real-world contexts. Also recognise when the normal approximation is appropriate and avoid common mistakes.

你应该掌握概率质量函数、均值与方差公式，并能将其应用于实际情境。同时要识别何时适合使用正态近似，并避免常见错误。

With regular practice on past paper questions, the binomial distribution will become a straightforward topic that earns you reliable marks in the IB exams.

通过定期练习历年真题，二项分布将成为IB考试中轻松得分的内容。

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IB Mathematics: Conditions and Typical Applications of Binomial Distribution | IB数学：二项分布的条件与典型应用