Exercise 8C: The Binomial Distribution | 练习8C：二项分布

📚 Exercise 8C: The Binomial Distribution | 练习8C：二项分布

The binomial distribution is one of the most important discrete probability distributions in the IB Mathematics curriculum. Whether you are following the Analysis and Approaches (AA) or Applications and Interpretation (AI) pathway, Exercise 8C typically challenges you to apply this model to situations with a fixed number of independent trials, each having the same probability of success. Mastering this exercise will sharpen your ability to calculate probabilities, use your GDC efficiently, and interpret means and variances in real-world contexts.

二项分布是IB数学课程中最重要的离散概率分布之一。无论你学习的是分析与方法（AA）还是应用与解释（AI），练习8C通常要求你将这一模型应用于具有固定次数独立试验、且每次成功概率相同的情形。掌握本练习将提升你计算概率、高效使用图形计算器以及解释实际问题中均值与方差的能力。

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

A binomial distribution arises from a binomial experiment: a process that produces exactly two outcomes, commonly labelled ‘success’ and ‘failure’, repeated a fixed number of times independently. The random variable X counts the number of successes in n trials. Because the distribution depends entirely on n and the constant success probability p, we write X ~ B(n, p).

二项分布源自二项试验：一个仅产生两种结果（通常标记为“成功”与“失败”）的过程，且该过程被独立地重复固定次数。随机变量 X 记录 n 次试验中成功的次数。由于该分布完全取决于 n 和恒定的成功概率 p，我们记作 X ~ B(n, p)。

In Exercise 8C, questions often start by asking you to recognise why a scenario fits a binomial model. Typical examples include the number of heads when flipping a coin ten times, the number of defective items in a batch, or students guessing correct answers on a multiple-choice test.

在练习8C中，题目通常先让你判断某个场景为何符合二项模型。常见的例子包括抛硬币十次出现正面的次数、一批产品中的次品数，或学生在选择题测试中猜对的题数。

2. Conditions for a Binomial Model | 二项模型的条件

To apply the binomial distribution, four strict conditions must hold. First, the number of trials n is fixed in advance. Second, each trial results in one of two mutually exclusive outcomes. Third, the probability of success p remains constant from trial to trial. Fourth, trials are independent — the outcome of one does not influence another.

要应用二项分布，必须满足四个严格条件。第一，试验次数 n 事先固定。第二，每次试验只产生两种互斥结果之一。第三，成功的概率 p 在每次试验中保持恒定。第四，各次试验相互独立——一次试验的结果不影响其他试验。

In IB exam questions, you may be asked to justify the use of a binomial distribution or to spot why a situation does not fit (e.g., sampling without replacement from a small population changes p). Be prepared to comment on each condition concisely.

在IB考题中，你可能需要说明为何使用二项分布，或指出某个情形不符合的原因（例如，从小总体不放回抽样会改变 p）。请准备好简明地评论每一个条件。

3. Notation and Parameters | 符号与参数

The binomial distribution has two parameters: n (number of trials) and p (probability of success). We often denote the failure probability as q = 1 – p. The notation X ~ B(n, p) is standard, and the probability that X equals exactly k successes is written as P(X = k).

二项分布有两个参数：n（试验次数）和 p（成功概率）。我们常将失败概率记为 q = 1 – p。标准记号是 X ~ B(n, p)，而 X 恰好等于 k 次成功的概率写作 P(X = k)。

You will encounter cumulative probabilities P(X ≤ k) and P(X ≥ k) frequently. Understanding the difference between ‘at most’, ‘at least’, ‘more than’ and ‘less than’ is crucial for using the correct GDC function or formula.

你会频繁遇到累积概率 P(X ≤ k) 和 P(X ≥ k)。理解“至多”“至少”“多于”“少于”之间的差异，对于使用正确的GDC函数或公式至关重要。

4. The Binomial Probability Formula | 二项概率公式

The probability of obtaining exactly k successes in n trials is given by:

恰好获得 k 次成功的概率由下式给出：

P(X = k) = ⁿCₖ pᵏ (1 – p)ⁿ⁻ᵏ

Here, ⁿCₖ (also read ‘n choose k’) is the binomial coefficient, defined as n! / [k!(n – k)!], and it counts the number of ways to arrange k successes among n trials.

其中，ⁿCₖ（也读作“n选k”）是二项式系数，定义为 n! / [k!(n – k)!]，它计算的是在 n 次试验中安排 k 次成功的方式数。

In Exercise 8C you may need to compute probabilities by hand for small n or when a GDC is not permitted. Remember that 0! = 1 and that ⁿC₀ = ⁿCₙ = 1.

在练习8C中，当 n 较小或不允许使用GDC时，你可能需要手算概率。请记住 0! = 1，并且 ⁿC₀ = ⁿCₙ = 1。

5. Calculating Binomial Probabilities Step by Step | 逐步计算二项概率

Let X ~ B(5, 0.4). To find P(X = 2), identify n = 5, p = 0.4, k = 2. The coefficient ⁵C₂ = 5!/(2!3!) = 10. Then P(X = 2) = 10 × (0.4)² × (0.6)³ = 10 × 0.16 × 0.216 = 0.3456.

设 X ~ B(5, 0.4)。要计算 P(X = 2)，先确定 n = 5，p = 0.4，k = 2。系数 ⁵C₂ = 5!/(2!3!) = 10。然后 P(X = 2) = 10 × (0.4)² × (0.6)³ = 10 × 0.16 × 0.216 = 0.3456。

When using the formula, always check that the powers correspond to the number of successes k and failures n−k. A common slip is to swap p and (1−p) or to forget the binomial coefficient entirely.

使用公式时，务必检查指数是否与成功次数 k 和失败次数 n−k 对应。常见的失误是交换 p 和 (1−p)，或完全遗漏二项式系数。

6. Using Your GDC for Binomial Probabilities | 使用图形计算器计算二项概率

Most IB papers expect efficient use of the GDC’s binomial functions. For P(X = k), use binompdf(n, p, k). For cumulative probabilities such as P(X ≤ k), use binomcdf(n, p, k). These are found under the DISTR menu on TI-84 or in the Calculator app on TI-Nspire.

大多数IB考试期望你高效使用GDC的二项函数。对于 P(X = k)，可用 binompdf(n, p, k)；对于累积概率如 P(X ≤ k)，可用 binomcdf(n, p, k)。这些函数可在TI-84的DISTR菜单或TI-Nspire的计算器应用中找到。

To find P(X ≥ 3) when n = 10, p = 0.2, rewrite it as 1 − P(X ≤ 2) and apply binomcdf(10, 0.2, 2). Always transform ‘greater than’ or ‘at least’ problems into a subtraction from 1, unless your GDC offers a direct complement function.

当 n = 10, p = 0.2 时，要求 P(X ≥ 3)，可改写为 1 − P(X ≤ 2) 并使用 binomcdf(10, 0.2, 2)。请始终将“大于”或“至少”问题转化为从 1 中减去，除非你的GDC提供直接的补函数。

7. Mean and Variance of a Binomial Distribution | 二项分布的均值与方差

The binomial distribution has simple formulas for its mean and variance. If X ~ B(n, p), then:

二项分布的均值与方差有简洁的公式。若 X ~ B(n, p)，则：

μ = E(X) = np

σ² = Var(X) = np(1 − p)

The standard deviation is σ = √[np(1 − p)]. These are derived from the fact that a binomial variable is the sum of n independent Bernoulli trials, each with mean p and variance p(1 − p).

标准差为 σ = √[np(1 − p)]。这些公式源于二项变量是 n 个独立伯努利试验之和，每个试验的均值为 p、方差为 p(1 − p)。

In Exercise 8C, you may need to use these results to solve for n or p or to compare theoretical predictions with observed data. For instance, if X ~ B(20, 0.3), the expected number of successes is 6, with variance 4.2.

在练习8C中，你可能需要运用这些结果求解 n 或 p，或将理论预测与观测数据进行比较。例如，若 X ~ B(20, 0.3)，则成功次数的期望值为 6，方差为 4.2。

8. Worked Examples: Applying the Binomial Distribution | 实例演练：应用二项分布

Example 1: A fair die is rolled 8 times. Find the probability of getting a ‘6’ exactly twice.

示例1：一枚均匀的骰子掷8次。求恰好出现2次“6”的概率。

Here n = 8, p = 1/6, k = 2. Using the GDC: binompdf(8, 1/6, 2) ≈ 0.260. Alternatively, ⁸C₂ (1/6)² (5/6)⁶ = 28 × (1/36) × (15625/46656) ≈ 0.260.

此处 n = 8，p = 1/6，k = 2。使用GDC：binompdf(8, 1/6, 2) ≈ 0.260。也可手算：⁸C₂ (1/6)² (5/6)⁶ = 28 × (1/36) × (15625/46656) ≈ 0.260。

Example 2: In a multiple-choice test with 15 questions, each having 4 options, a student guesses all answers. Find the probability of scoring at least 10 correct.

示例2：一次有15道题的选择题测试，每题4个选项，一名学生全凭猜测作答。求猜对至少10题的概率。

n = 15, p = 0.25. We need P(X ≥ 10) = 1 − P(X ≤ 9). GDC: 1 − binomcdf(15, 0.25, 9) ≈ 1 − 0.9996 = 0.0004. The chance is extremely small, which highlights how unlikely guessing high scores is.

n = 15，p = 0.25。需求 P(X ≥ 10) = 1 − P(X ≤ 9)。GDC：1 − binomcdf(15, 0.25, 9) ≈ 1 − 0.9996 = 0.0004。此概率极小，突显了纯粹猜测得高分的可能性有多低。

9. Shape of the Binomial Distribution | 二项分布的形状

The shape of a binomial distribution depends on n and p. When p = 0.5, the distribution is perfectly symmetric. For p < 0.5, it is skewed to the right (positively skewed); for p > 0.5, it is skewed to the left. As n increases, the distribution becomes more symmetric and bell-shaped, eventually approximating a normal distribution.

二项分布的形状取决于 n 和 p。当 p = 0.5 时，分布完全对称。当 p < 0.5 时，分布右偏（正偏）；当 p > 0.5 时，分布左偏。随着 n 增大，分布变得更加对称并呈钟形，最终逼近正态分布。

You may be asked to sketch probability histograms for small n, labelling the probability of each outcome. Understanding the shape helps in checking whether a normal approximation is appropriate.

你可能会被要求绘制 n 较小时的概率直方图，并标出每个结果的概率。理解形状有助于判断正态近似是否合适。

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

When n is large enough that both np ≥ 10 and n(1 − p) ≥ 10, X ~ B(n, p) can be approximated by a normal distribution N(np, np(1 − p)). A continuity correction is usually applied to improve accuracy: for P(X ≤ k), use the normal probability up to k + 0.5.

当 n 足够大，使得 np ≥ 10 和 n(1 − p) ≥ 10 同时满足时，X ~ B(n, p) 可用正态分布 N(np, np(1 − p)) 近似。通常会应用连续性校正以提高精度：对于 P(X ≤ k)，使用截至 k + 0.5 的正态概率。

In IB AA HL, you might need to find approximate probabilities without a GDC. For example, if 200 coins are tossed, the number of heads is approximately N(100, 50). P(X ≤ 110) with continuity correction becomes P(Y < 110.5), where Y ~ N(100, 50).

在IB AA HL课程中，你可能需要在不使用GDC的情况下求近似概率。例如，若抛掷200枚硬币，正面次数近似为 N(100, 50)。带连续性校正的 P(X ≤ 110) 变为 P(Y < 110.5)，其中 Y ~ N(100, 50)。

11. Common Mistakes and How to Avoid Them | 常见错误与如何避免

One frequent mistake is confusing binompdf and binomcdf on the GDC. Remember: ‘pdf’ gives the probability of exactly one value, while ‘cdf’ accumulates all probabilities up to a value. Always double-check whether the question asks for P(X = k) or P(X ≤ k).

一个常见错误是在GDC上混淆 binompdf 和 binomcdf。记住：’pdf’ 给出恰好一个值的概率，而 ‘cdf’ 则累加至某个值的所有概率。务必反复确认题目问的是 P(X = k) 还是 P(X ≤ k)。

Another pitfall is misinterpreting ‘at least’ or ‘more than’. For instance, P(X ≥ 4) is not binomcdf(10, p, 4); it must be 1 − P(X ≤ 3). Missing the complement can lead to answers that are off by a large margin.

另一个陷阱是误解“至少”或“多于”。例如，P(X ≥ 4) 不是 binomcdf(10, p, 4)；而应该是 1 − P(X ≤ 3)。遗漏补数可能导致答案出现巨大偏差。

Also, avoid assuming that probability stays constant when sampling without replacement unless the population is very large. The binomial model applies only when p remains unchanged from trial to trial.

此外，除非总体非常大，否则不要在不放回抽样时假定概率恒定。二项模型仅当 p 在每次试验间保持不变时才适用。

12. Summary and Key Takeaways | 总结与要点

Exercise 8C consolidates your ability to recognise, apply, and interpret the binomial distribution. Key formulas: P(X = k) = ⁿCₖ pᵏ (1 − p)ⁿ⁻ᵏ, mean np, variance np(1 − p). GDC functions binompdf and binomcdf are essential tools for efficiency and accuracy.

练习8C巩固了你识别、应用和解释二项分布的能力。关键公式：P(X = k) = ⁿCₖ pᵏ (1 – p)ⁿ⁻ᵏ，均值 np，方差 np(1 − p)。GDC函数 binompdf 和 binomcdf 是保证高效和准确的基本工具。

Always verify the four binomial conditions before applying the model. For skewed distributions, remember that the normal approximation requires both np ≥ 10 and n(1 − p) ≥ 10, and use continuity correction for discrete-to-continuous approximations. Practice transforming ‘at least’ and ‘at most’ phrases into the correct cumulative forms.

在应用模型之前，务必核实四个二项条件。对于偏态分布，记住正态近似要求 np ≥ 10 和 n(1 − p) ≥ 10 同时满足，并在离散到连续近似时使用连续性校正。练习将“至少”和“至多”等表述转化为正确的累积形式。

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