📚 Exercise 15B.2: Binomial Distribution Problem Solving | 练习15B.2：二项分布问题求解
In IB Mathematics, Exercise 15B.2 typically focuses on applying the binomial distribution to real-world scenarios. This set of problems challenges students to identify the parameters n and p, calculate probabilities of exactly k successes, and use cumulative probabilities for ranges of outcomes. Mastering these skills is essential for success in both the Analysis & Approaches and Applications & Interpretation courses. This article breaks down the key methods, common pitfalls, and step-by-step solutions to help you tackle Exercise 15B.2 with confidence.
在IB数学中，练习15B.2通常侧重于将二项分布应用于实际场景。这组题目要求学生识别参数n和p，计算恰好k次成功的概率，并使用累积概率处理结果的范围。掌握这些技能对于在分析与方法和应用与解释两门课程中取得成功至关重要。本文分解了关键方法、常见陷阱和逐步解法，帮助你自信应对练习15B.2。
1. Recognizing a Binomial Setting | 识别二项分布的条件
Before diving into the exercises, always verify that a situation satisfies the four binomial conditions: a fixed number of trials (n), each trial has only two outcomes (success/failure), the probability of success (p) remains constant, and trials are independent. Exercise 15B.2 often presents word problems where these assumptions must be checked implicitly.
在深入练习之前，务必验证情境满足四个二项条件：固定次数的试验（n），每次试验只有两种结果（成功/失败），成功概率（p）保持不变，且各次试验相互独立。练习15B.2常给出应用题，需要你隐含地验证这些假设。
2. Defining Parameters n and p | 定义参数 n 和 p
For each sub‑question in 15B.2, extract the total number of trials n and the probability of a single success p from the context. Remember that p can be given directly (e.g., 20% defective items) or inferred from a long‑run relative frequency. In IB exams, p is often expressed as a decimal, so convert percentages accordingly.
对于15B.2的每个子问题，从上下文中提取试验总次数n和单次成功的概率p。记住p可以直接给出（例如20%的次品率），也可以从长期相对频率中推断。在IB考试中，p通常以小数表示，因此需要相应转换百分比。
3. Using the Binomial Probability Formula | 使用二项概率公式
robability of obtaining exactly x successes in n trials is given by P(X = x) = C(n, x) · px · (1-p)n-x, where C(n, x) is the binomial coefficient n!/(x!(n-x)!). Exercise 15B.2 often asks for 'exactly' scenarios. Show all working: first the combination, then the powers, then the product.
在n次试验中获得恰好x次成功的概率由P(X=x) = C(n,x) · px · (1-p)n-x给出，其中C(n,x)是二项式系数 n!/(x!(n-x)!)。练习15B.2常要求计算“恰好”情境的概率。展示完整步骤：先写组合数，再写幂次，最后乘出结果。
P(X = x) = ⁿCₓ · pˣ · (1-p)ⁿ⁻ˣ
4. Calculating Cumulative Probabilities | 计算累积概率
Many 15B.2 problems require P(X ≤ k) or P(X ≥ k). Use your GDC' P(X ≤ k), and for P(X ≥ k), rewrite it as 1 − P(X ≤ k−1). Be careful with the boundaries: 'at least 5' X ≥ 5, which translates to 1 − P(X ≤ 4). Writing the inequality explicitly reduces errors.
许多15B.2题目要求计算P(X ≤ k)或P(X ≥ k)。使用图形计算器的binomcdf函数求P(X ≤ k)，而对于P(X ≥ k)，改写为 1 − P(X ≤ k−1)。注意边界：“至少5次”对应X ≥ 5，转化为 1 − P(X ≤ 4)。明确写出不等式可减少错误。
5. Working with 'More Than' and 'Less Than' | 处理“多于”和“少于”
Phrases like 'more than 3' (X > 3) mean X ≥ 4, so P(X > 3) = 1 − P(X ≤ 3). Similarly, 'fewer than 2' (X < 2) means X ≤ 1. Exercise 15B.2 deliberately mixes strict and non‑ t this understanding. Always restate the inequality in terms of ≤ before applying the GDC.
像“多于3次”(X > 3)这样的短语意味着 X ≥ 4，因此P(X > 3) = 1 − P(X ≤ 3)。类似地，“少于2次”(X < 2)意味着 X ≤ 1。练习15B.2故意混合严格与非严格不等式来检验这种理解。务必在使用计算器前将不等式用≤形式重新表述。
6. Using Tables for Range Probabilities | 使用表格求区间概率
a−1). Drawing a small table of x and P(X = x) is a helpful visual check, especially in non‑calculator sections. A typical table for a 15B.2 exercise might look like:
对于求P(a ≤ X ≤ b)的问题，既可以逐个相加P(X = x)的值，也可以使用累积分布：P(X ≤ b) − P(X ≤ a−1)。绘制一个包含x和P(X=x)的小表格是一种有用的视觉检查，尤其在非计算器部分。练习15B.2中一个典型的表格可能如下：
| x | P(X = x) |
|---|---|
| 0 | 0.0008 |
| 1 | 0.0064 |
| 2 | 0.0512 |
| 3 | 0.2048 |
| 4 | 0.4096 |
| 5 | 0.3277 |
7. Solving for an Unknown Number of Trials | 求解未知试验次数
that P(X ≥ 1) > 0.99. This requires using the complement and logarithms. Set up 1 − (1−p)n > 0.99, then (1−p)n < 0.01. Taking logs, n > log(0.01)/log(1−p). Round up to the next integer.
练习15B.2偶尔会包含逆向问题：给定一个概率，求n。例如，求最小的n使得P(X ≥ 1) > 0.99。这需要使用补集和对数。建立不等式 1 − (1−p)n > 0.99，然后 (1−p)n < 0.01。取对数得 n > log(0.01)/log(1−p)。向上取整到下一个整数。
n ≥ ⌈ log(1 − target) / log(1−p) ⌉
8. Real‑World Interpretation and Communication | 现实世界解读与表述
IB examiners expect you to interpret the result in context. After computing a probability, write a sentence like 'There is a 12.5% chance that exactly 2 out of 10 light bulbs will be defective.' Exercise 15B.2 often awards marks for this interpretation. Avoid generic phrasing and tie your answer back to the specific scenario given.
IB考官期望你在上下文中解读结果。计算概率后，写出诸如“10个灯泡中有2个次品的概率为12.5%”的句子。练习15B.2常为这种解读赋分。避免泛泛的表述，将答案与所给的具体情境联系起来。
9. Checking Answer Reasonableness | 检查答案的合理性
like p=0.5 and small n, to verify your calculator or manual calculations during revision.
二项概率值应始终介于0和1之间。快速合理性检查：所有可能x的P(X=x)之和必须等于1。如果你的答案超过1或为负数，则出错了。利用简单的基准示例，例如p=0.5和小的n，在复习时验证你的计算器或手算结果。
10. Common Errors in Exercise 15B.2 | 练习15B.2中的常见错误
Students often confuse the number of successes with the number of failures. Another frequent mistake is using binompdf when binomcdf is required, or vice‑versa. Misreading 'at least' as 'more than' can also cost marks. Keep a checklist: identify the keyword, translate it to an inequality, decide which calculator function to use, and finally interpret the output.
学生经常混淆成功次数与失败次数。另一个常见错误是需要使用binomcdf时却用了binompdf，或反之。将“至少”误读为“多于”也会丢分。保持一份检查清单：识别关键词，转化为不等式，决定使用哪个计算器函数，最后解读输出结果。
11. Practising with Varied Scenarios | 在不同情境中练习
actise identifying n and p in each setting because the wording can be subtle. For example, '10% of patients experience side effects' gives p=0.1, and 'a sample of 20 patients' gives n=20. Repeated exposure builds fluency.
练习15B.2涵盖质量控制、医学检测、民意调查和机会游戏等情境。练习在每种情境中识别n和p，因为措辞可能很微妙。例如，“10%的患者出现副作用”给出p=0.1，“一个20名患者的样本”给出n=20。反复接触可培养熟练度。
12. Exam‑Style Strategy for Binomial Questions | 二项题目的考试策略
In an IB exam, show your method: state the distribution as X ~ command and result. Round probabilities to three significant figures unless instructed otherwise. Manage your time by using your GDC efficiently for cumulative probabilities—manual calculation is only expected when n is small and a GDC is not allowed.
在IB考试中，要展示方法：写出分布形式 X ~ B(n, p)，写出概率语句（例如 P(X ≤ 2)），并给出计算器命令和结果。除非另有指示，将概率四舍五入到三位有效数字。通过高效使用计算器处理累积概率来管理时间——仅当n较小且不允许使用计算器时才要求手算。
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
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