Mastering Exercise 12B.3: Discrete Random Variables & Binomial Distributions | 掌握练习12B.3：离散随机变量与二项分布

📚 Mastering Exercise 12B.3: Discrete Random Variables & Binomial Distributions | 掌握练习12B.3：离散随机变量与二项分布

Exercise 12B.3 in the IB Mathematics curriculum challenges students to apply discrete random variable concepts and binomial distributions to real-world-like problems. This article breaks down the key ideas, from probability distribution tables to expected value, variance, and the binomial formula, providing worked examples and common pitfalls to help you master the exercise with confidence.

IB 数学课程中的练习 12B.3 要求学生将离散随机变量的概念与二项分布应用到贴近实际的问题中。本文拆解核心知识点，从概率分布表到期望值、方差和二项公式，提供例题讲解与常见错误分析，帮助你自信掌握这一练习。

1. Understanding Discrete Random Variables | 理解离散随机变量

A discrete random variable (DRV) takes a countable number of distinct values, each with an associated probability. In IB problems, the variable X often represents outcomes like the score on a die, the number of successes in trials, or the net gain in a game.

离散随机变量（DRV）取可数个不同值，每个值对应一个概率。在 IB 题目中，变量 X 常表示诸如骰子点数、试验成功次数或游戏净收益等结果。

The sum of all probabilities for a DRV must equal exactly 1. This is your first check when constructing or verifying a probability distribution.

DRV 所有概率之和必须严格等于 1。这是你构建或验证概率分布时首先要检查的内容。

2. Probability Distribution Tables | 概率分布表

A distribution table lists each possible value x of X together with P(X = x). Ensure the table is complete: missing a value can lead to incorrect expectations or variances.

分布表列出 X 的每个可能取值 x 以及对应的 P(X = x)。确保表格完整：遗漏某个值会导致期望值或方差计算错误。

x	1	2	3
P(X=x)	0.3	0.5	0.2

In Exercise 12B.3, you may need to find missing probabilities using the sum-to-one rule or apply it to calculate expected values.

在练习 12B.3 中，你可能需要用概率之和为 1 的规则求出缺失的概率，或应用它来计算期望值。

3. Expected Value (Mean) | 期望值（均值）

The expected value E(X) is the long-run average outcome and is calculated as:

期望值 E(X) 是长期平均结果，计算公式为：

E(X) = Σ x · P(X = x)

Multiply each value by its probability and sum the results. This gives the theoretical mean of the distribution.

将每个值乘以其概率，再将结果相加。这样可得到分布的理论均值。

For the table above: E(X) = 1×0.3 + 2×0.5 + 3×0.2 = 1.9. Always show the substitution step clearly to gain method marks in IB exams.

对于上表：E(X) = 1×0.3 + 2×0.5 + 3×0.2 = 1.9。在 IB 考试中，一定要清晰展示代入步骤以获得方法分。

4. Variance and Standard Deviation | 方差与标准差

Variance measures the spread of the distribution. The formula preferred in IB is:

方差衡量分布的离散程度。IB 考试中推荐使用的公式为：

Var(X) = E(X²) − [E(X)]²

First compute E(X²) = Σ x² · P(X = x), then subtract the square of the mean. Standard deviation is the square root of variance: σ = √Var(X).

先计算 E(X²) = Σ x² · P(X = x)，然后减去均值的平方。标准差是方差的平方根：σ = √Var(X)。

Using the previous example: E(X²) = 1²×0.3 + 2²×0.5 + 3²×0.2 = 0.3 + 2.0 + 1.8 = 4.1. Then Var(X) = 4.1 − (1.9)² = 4.1 − 3.61 = 0.49, so σ = 0.7.

沿用前例：E(X²) = 1²×0.3 + 2²×0.5 + 3²×0.2 = 0.3 + 2.0 + 1.8 = 4.1。然后 Var(X) = 4.1 − (1.9)² = 4.1 − 3.61 = 0.49，因此 σ = 0.7。

Always interpret: a larger variance means outcomes are more spread out from the mean.

始终记住：方差越大，意味着结果相对于均值的离散程度越大。

5. Linear Transformations of Random Variables | 随机变量的线性变换

When a random variable is transformed by Y = aX + b, the expectation and variance change in predictable ways:

当随机变量通过 Y = aX + b 进行变换时，期望与方差的变化规律如下：

E(Y) = a E(X) + b

Var(Y) = a² Var(X)

Note that adding a constant b shifts the mean but does not affect variance or standard deviation. Multiplying by a scales the mean by a and scales the variance by a².

注意，加常数 b 会平移均值，但不影响方差或标准差。乘以系数 a 会将均值缩放 a 倍，并将方差缩放 a² 倍。

Exercise 12B.3 often includes problems involving net profit or cost functions, where you apply these rules to find the new mean and standard deviation.

练习 12B.3 常包含涉及净利润或成本函数的问题，此时你需要运用这些规则求出新的均值和标准差。

6. Introduction to Binomial Distribution | 二项分布简介

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success p. It is denoted X ~ B(n, p), where n is the number of trials.

二项分布描述的是在固定次数的独立试验中成功的次数，每次试验成功的概率相同，均为 p。记作 X ~ B(n, p)，其中 n 为试验次数。

This distribution belongs to the discrete family, but its probabilities follow a specific formula rather than being given in a table. When you see phrases like ‘number of heads in 10 tosses’ or ‘number of defective items in a sample’, think binomial.

该分布属于离散型分布，但其概率遵循特定公式，而不是直接给出表格。当你看到如“投掷 10 次硬币正面的次数”或“样本中次品的数量”等表述时，就应考虑二项分布。

7. Conditions for a Binomial Experiment | 二项试验的条件

For a variable to be modelled by a binomial distribution, four conditions must hold:

要使一个变量能用二项分布建模，必须满足四个条件：

Fixed number of trials n: The experiment runs exactly n times.
Fixed number of trials n: 试验次数固定，恰好进行 n 次。
Independent trials: The outcome of one trial does not affect another.
Independent trials: 各次试验相互独立，一次试验的结果不影响其他试验。
Two outcomes per trial: Often called ‘success’ and ‘failure’.
每次试验仅两个结果： 通常称为“成功”与“失败”。
Constant probability of success p: The same p applies to every trial.
成功概率 p 恒定： 每次试验的成功概率不变。

Exercise questions may ask you to identify whether a scenario fits a binomial model; always check these four points explicitly.

练习题可能会要求你判断某个情景是否适合二项模型；务必逐一核查这四点。

8. Binomial Probability Formula | 二项概率公式

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

若 X ~ B(n, p)，则恰好取得 k 次成功的概率为：

P(X = k) = nC_k · p^k · (1 − p)^n−k

Here nC_k = n! / (k!(n−k)!) is the binomial coefficient, found on your calculator. The term p^k represents the probability of k successes, and (1 − p)^n−k is the probability of the remaining trials being failures.

这里 nC_k = n! / (k!(n−k)!) 为二项式系数，可在计算器上求得。p^k 表示 k 次成功的概率，(1 − p)^n−k 则是其余试验失败的概率。

When using a GDC, you can use binompdf(n, p, k) for exact probability. For cumulative probabilities, use binomcdf(n, p, k) which gives P(X ≤ k).

使用图形计算器时，可用 binompdf(n, p, k) 计算精确概率，用 binomcdf(n, p, k) 计算累积概率 P(X ≤ k)。

9. Worked Example: Expected Value & Variance of Binomial | 例题：二项分布的期望与方差

A standard IB problem: A fair die is rolled 12 times. Let X be the number of times a ‘six’ appears. Find E(X) and Var(X).

一道典型 IB 题目：一枚公平的骰子被投掷 12 次。设 X 为出现“6 点”的次数。求 E(X) 和 Var(X)。

Here, n = 12, p = 1/6. So X ~ B(12, 1/6). The formulas for binomial distributions are:

这里 n = 12，p = 1/6。因此 X ~ B(12, 1/6)。二项分布的公式为：

E(X) = n p

Var(X) = n p (1 − p)

Substitute: E(X) = 12 × (1/6) = 2. Var(X) = 12 × (1/6) × (5/6) = 12 × 5/36 = 60/36 = 5/3 ≈ 1.667. Standard deviation is √(5/3) ≈ 1.29.

代入计算：E(X) = 12 × (1/6) = 2。Var(X) = 12 × (1/6) × (5/6) = 12 × 5/36 = 60/36 = 5/3 ≈ 1.667。标准差为 √(5/3) ≈ 1.29。

Remember that these shortcut formulas only apply to binomial random variables, not to general discrete distributions.

请记住，这些简化公式仅适用于二项随机变量，不适用于一般的离散分布。

10. Solving Exercise 12B.3-Style Problems | 解答练习 12B.3 型题目

Exercise 12B.3 typically blends several concepts: you might first need to verify that a situation is binomial, then compute individual probabilities, and finally find the expected number of successes or the probability of a range of outcomes.

练习 12B.3 通常混合多个概念：你可能需要先验证某情景符合二项分布，然后计算个别概率，最后求成功次数的期望值或某个结果范围的概率。

A common type of question gives a probability distribution table for a DRV, asks for a missing value, then asks for E(X) and Var(X), and finally applies a linear transformation for cost or profit. Practise moving from one step to the next smoothly.

一类常见题目是给出一个 DRV 的概率分布表，要求补全缺失值，再求 E(X) 和 Var(X)，最后对成本或收益进行线性变换。应练习顺畅地从前一步过渡到下一步。

When dealing with binomial scenarios, always state the distribution clearly: ‘X ~ B(n, p)’ and specify the values of n and p before performing calculations.

处理二项分布情景时，务必先清晰写出分布：’X ~ B(n, p)’，并在计算前明确给出 n 和 p 的值。

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

Misidentifying the distribution: Not all success-failure situations are binomial. If p changes or trials are not independent, use a different model.

错误判识别分布类型：并非所有成功-失败型情景都符合二项分布。如果 p 发生变化或试验不独立，应使用其他模型。

Forgetting the complement: P(X ≥ k) is often easier as 1 − P(X ≤ k−1), especially with GDC binomcdf.

忘记使用补集：P(X ≥ k) 通常可转化为 1 − P(X ≤ k−1) 来计算，尤其借助 GDC 的 binomcdf 功能时更为简便。

Variance misinterpretation: When finding standard deviation, always take the square root at the end. Variance units are squared units of X.

对方差的理解偏差：在求标准差时，务必在最后一步取平方根。方差的单位是 X 单位的平方。

Practice writing intermediate steps clearly; IB examiners award marks for correct method even if the final answer slips. Also, double-check that probabilities sum to 1 before moving on.

练习清晰书写中间步骤；IB 考官对正确方法给予步骤分，即使最终答案有误。此外，在继续计算前务必核查概率之和为 1。

12. Connecting to the Bigger Picture | 联系更广的知识体系

Mastering Exercise 12B.3 builds a strong foundation for later topics such as the normal approximation to the binomial, hypothesis testing, and statistical applications in the IB internal assessment. The ability to handle DRVs with confidence will also support your work in continuous distributions.

掌握练习 12B.3 可为后续知识打下坚实基础，例如二项分布的正态近似、假设检验以及 IB 内部评估中的统计应用。自信地处理 DRV 的能力也会助力你对连续分布的学习。

Return to these fundamentals whenever you encounter more complex probability models; the logic of expected value and variance remains the same across the syllabus.

每当你遇到更复杂的概率模型时，都可以回到这些基础原理；期望值与方差的逻辑在整个课程大纲中贯穿始终。

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Mastering Exercise 12B.3: Discrete Random Variables & Binomial Distributions | 掌握练习12B.3：离散随机变量与二项分布