📚 Exercise 13C.2: Binomial Distribution Practice | 练习13C.2：二项分布练习
This article provides a complete walkthrough for IB Mathematics Exercise 13C.2, focusing on binomial distribution problems. By working through these examples and explanations, you will learn how to identify binomial experiments, calculate probabilities using the binomial formula, find expected values and variances, and apply these skills in examination-style questions. Use this guide to deepen your understanding and check your own answers.
本文为 IB 数学练习 13C.2 提供完整的讲解，重点讨论二项分布问题。通过这些例题和解释，您将学会如何识别二项试验、使用二项公式计算概率、求期望值和方差，并将这些技能应用于考试型题目。请用这份指南加深理解并核对您的答案。
1. Recognising a Binomial Experiment | 识别二项试验
A binomial experiment arises when a fixed number of independent trials are performed, each having exactly two possible outcomes (success or failure), and the probability of success remains constant across trials. Common examples include flipping a fair coin multiple times, checking defective items in a quality control sample, or asking yes/no survey questions.
当进行固定次数的独立试验，每次试验只有两种可能结果（成功或失败），且每次试验的成功概率保持不变时，就产生了二项试验。常见例子包括多次抛掷公平硬币、在质量控制的样本中检验缺陷品，或进行是/否的问卷调查。
To confirm you are dealing with a binomial situation, check these four conditions: there are n identical trials; each trial results in one of two outcomes; the probability of success p is the same for every trial; and trials are independent. If any condition is violated, the binomial model may not apply.
要确认您处理的是二项分布情形，请检查以下四个条件：共有 n 次相同试验；每次试验只有两种结果之一；每次试验的成功概率 p 相同；且各次试验相互独立。如果违反任一条件，二项模型便不适用。
2. The Binomial Probability Formula | 二项概率公式
The probability of obtaining exactly k successes in n independent trials is given by:
在 n 次独立试验中恰好获得 k 次成功的概率由下式给出：
P(X = k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ
where C(n, k) is the binomial coefficient, also written as ⁿCₖ or (n choose k). It is calculated as n! / [k! (n − k)!].
其中 C(n, k) 是二项式系数，也写作 ⁿCₖ 或 (n choose k)，计算为 n! / [k! (n − k)!]。
This formula combines three parts: the number of ways to choose which trials are successes, the probability of those successes, and the probability of the remaining failures. When solving Exercise 13C.2, carefully substitute the values of n, k, and p to evaluate the exact probability.
该公式由三部分组成：选择哪些试验成为成功的方式数、这些成功发生的概率，以及剩余失败的概率。在求解练习 13C.2 时，请仔细代入 n、k 和 p 的值以计算确切概率。
3. Using the Formula – Worked Example | 使用公式 – 解题示例
o sixes. Here n = 5, p = 1/6, and k = 2. First compute the binomial coefficient C(5,2) = 5!/(2!3!) = 10. Then P(X = 2) = 10 · (1/6)² · (5/6)³ ≈ 0.1608.
假设一个公平的骰子投掷 5 次，我们要求恰好掷出两个 6 点的概率。这里 n = 5，p = 1/6，k = 2。首先计算二项式系数 C(5,2) = 5!/(2!3!) = 10。然后 P(X = 2) = 10 · (1/6)² · (5/6)³ ≈ 0.1608。
Always show the setup step by step in your working. In an IB exam, marks are awarded for identifying n, p, k and for substituting correctly, even if the final numerical answer contains a minor arithmetic slip.
在解题过程中，务必逐步展示步骤。在 IB 考试中，即使最终数值答案有小的算术失误，正确指出 n、p、k 并正确代入公式也会获得分数。
4. Cumulative Probabilities | 累积概率
Questions often ask for the probability of at most r successes or at least r successes. For 'at most 2' successes, we sum P(X = 0) + P(X = 1) + P(X = 2). For 'at least 2', we can sum from k = 2 to n, or use the complement rule: P(X ≥ 2) = 1 − P(X ≤ 1).
问题常要求计算至多 r 次成功或至少 r 次成功的概率。对于「至多 2 次」成功，我们将 P(X = 0) + P(X = 1) + P(X = 2) 求和。对于「至少 2 次」，可以从 k = 2 到 n 求和，或使用补集规则：P(X ≥ 2) = 1 − P(X ≤ 1)。
When n is large, manual summation is tedious. In such cases, IB expects you to use the binomial cumulative distribution function on your graphical display calculator (GDC). Be sure to know how to access binomCdf on your TI-Nspire or Casio model.
当 n 很大时，手动求和十分繁琐。这种情况下，IB 期望您使用图形计算器 (GDC) 上的二项累积分布函数。请确保您知道如何在 TI-Nspire 或 Casio 机型上使用 binomCdf 功能。
5. Expected Value and Variance of a Binomial Distribution | 二项分布的期望值与方差
For a binomial random variable X ~ ). These formulas are simple yet powerful for checking the reasonableness of your answers.
对于二项随机变量 X ~ B(n, p)，成功的期望次数为 E(X) = np，方差为 Var(X) = np(1 − p)。这些公式简单而强大，可用于检查答案的合理性。
ability result that suggests an extremely different outcome might signal a calculation error.
例如，如果您投掷骰子 120 次并数出 6 点的次数，平均期望约有 120 × (1/6) = 20 次 6 点。若某个概率结果暗示了极其不同的结果，则可能表明存在计算错误。
6. Typical Exam Question Type 1 – Direct Calculation | 典型考试题型 1 – 直接计算
A question from Exercise 13C.2 might read: ' ead) = 0.3. The coin is tossed 8 times. Find the probability of obtaining exactly 3 heads.' Here n = 8, p = 0.3, k = 3. The solution is P(X = 3) = C(8,3) · (0.3)³ · (0.7)⁵.
练习 13C.2 中的一道题可能为：「一枚偏倚硬币的正面概率为 0.3。将此硬币抛掷 8 次。求恰好得到 3 次正面的概率。」这里 n = 8, p = 0.3, k = 3。解答为 P(X = 3) = C(8,3) · (0.3)³ · (0.7)⁵。
= 0.027, (0.7)⁵ ≈ 0.16807. Multiplying gives 56 × 0.027 × 0.16807 ≈ 0.254. So about a 25.4% chance. Writing intermediate steps earns method marks.
计算 C(8,3) = 56。接着 (0.3)³ = 0.027，(0.7)⁵ ≈ 0.16807。相乘得 56 × 0.027 × 0.16807 ≈ 0.254，即约 25.4% 的几率。写出中间步骤可赢得方法分。
7. Typical Exam Question Type 2 – Using Inequalities | 典型考试题型 2 – 使用不等式
Another common exercise asks: 'Find the probability of at least 4 heads.' =0), P(X=1), P(X=2), and P(X=3) using the binomial formula, sum them, and subtract from 1.
另一个常见练习题问道：「求至少 4 次正面的概率。」对于同一枚硬币，P(X ≥ 4) = 1 − P(X ≤ 3)。您需要使用二项公式计算 P(X=0)、P(X=1)、P(X=2) 和 P(X=3)，求和，再从 1 中减去。
mistakenly use P(X < 3). Always double-check whether the inequality includes equality. For 'more than 4', you need P(X ≥ 5).
许多学生在计算 P(X ≤ 3) 时会漏掉 P(X=3) 这一项；他们错误地计算了 P(X < 3)。请务必仔细检查不等式是否包含等号。对于「多于 4 次」，您需要的是 P(X ≥ 5)。
8. Using the GDC Efficiently | 高效使用图形计算器
r, upper). For P(X = 5) use binomPdf(10,0.5,5). For P(3 ≤ X ≤ 7) use binomCdf(10,0.5,3,7).
您的图形计算器可用 binomPdf(n, p, k) 计算单个概率，用 binomCdf(n, p, lower, upper) 计算累积概率。求 P(X = 5) 可使用 binomPdf(10,0.5,5)。求 P(3 ≤ X ≤ 7) 可使用 binomCdf(10,0.5,3,7)。
In the IB exam, you are expected to state the calculator command or indicate GDC use in your written solution. Simply writing the final answer without showing the method may lose marks, so always include a brief explanation.
在 IB 考试中，期望您在书面解答中注明计算器命令或指明使用了 GDC。仅写出最终答案而不展示方法可能会失分，因此始终要附上简要说明。
9. Common Pitfalls and How to Avoid Them | 常见陷阱及避免方法
One of the biggest mistakes in binomial problems is confusing p with 1−p. Always identify what you call a 'success' clearly and ensure you apply the correct probability throughout. Also, be careful not to misapply the complement rule when the inequality is strict.
二项问题中最大的错误之一是混淆 p 与 1−p。务必清楚地定义您所谓的「成功」，并确保全程使用正确的概率。同时，注意当不等式是严格不等时不要误用补集规则。
Another trap is forgetting that binomial trials must be independent. For example, picking cards from a deck without replacement changes probabilities, so the distribution is no longer binomial (it becomes hypergeometric). Always verify independence.
另一个陷阱是忘记二项试验必须相互独立。例如，从一副牌中不放回地抽牌会改变概率，因此分布不再属于二项分布（它变成了超几何分布）。请始终验证独立性。
10. Linking to the Binomial Theorem | 与二项式定理的联系
The binomial distribution formula is intimately connected to the binomial theorem from algebra. If you expand (p + (1−p))ⁿ using the binomial theorem, the general term is C(n,k) pᵏ (1−p)ⁿ⁻ᵏ, exactly matching the probability of k successes. This explains why the sum of all probabilities equals 1.
二项分布公式与代数中的二项式定理密切相关。如果用二项式定理展开 (p + (1−p))ⁿ，其通项为 C(n,k) pᵏ (1−p)ⁿ⁻ᵏ，恰好与 k 次成功的概率吻合。这就解释了为何所有概率之和等于 1。
Recognising this link helps reinforce the concept that the binomial coefficient counts the number of paths with k successes, while the powers represent the probability weight of each path. This is a beautiful bridge between combinatorics and probability.
认识到这种联系有助于强化概念：二项式系数计算了具有 k 次成功的路径数，而幂次代表了每条路径的概率权重。这是组合数学与概率论之间一座优美的桥梁。
11. Real-World Applications | 实际应用
Binomial distributions model many everyday phenomena: the number of defective products in a factory batch, the number of patients responding to a treatment in a clinical trial, or the number of voters supporting a particular policy in a poll. IB questions often frame problems in such realistic contexts.
二项分布为许多日常现象建模：一批产品中的次品数量、临床试验中对治疗有反应的患者人数，或民意调查中支持某项政策的选民人数。IB 考题常以这类现实情境为背景。
When interpreting results, remember that while the model gives probabilities, real-world conclusions require careful reasoning. A low probability does not necessarily prove a coin is biased; it merely suggests that the observed outcome would be unusual under a fair assumption.
在解释结果时，请记住，虽然模型给出了概率，但现实世界的结论需要谨慎推理。低概率并不一定证明硬币有偏倚；它仅表明在公平假设下观察到的结果是异常的。
12. Summary and Practice Tips for Exercise 13C.2 | 总结及练习 13C.2 的练习建议
To excel in Exercise 13C.2, master the four conditions for a binomial setting, memorise the probability formula, and practise calculating both individual and cumulative probabilities by hand and with a GDC. Work through all problems in the exercise, comparing your answers with worked solutions.
要想在练习 13C.2 中表现出色，请掌握二项分布场景的四个条件，熟记概率公式，并练习手动及使用 GDC 计算单项概率和累积概率。完成练习中的所有题目，并将您的答案与解答范例进行比较。
(np(1−p)). These are frequently tested and can be memorised quickly. With consistent practice, binomial distribution questions become a reliable source of marks.
制作一张包含公式 E(X) = np 和 Var(X) = np(1−p) 以及标准差 SD(X) = √(np(1−p)) 的摘要表。这些公式常被考到，且可快速记住。通过持续练习，二项分布题目将成为可靠的得分来源。
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