IB Math: Discrete Probability Distributions – Key Question Types | IB数学：离散概率分布题型梳理

📚 IB Math: Discrete Probability Distributions – Key Question Types | IB数学：离散概率分布题型梳理

Discrete probability distributions are a core topic in IB Mathematics, appearing in both Analysis and Approaches (AA) and Applications and Interpretation (AI). Examiners love to test whether you can construct a probability distribution table, calculate expected values, and apply named distributions such as binomial and Poisson. This article organises the question types by skill, from basic definitions to GDC-heavy problems.

离散概率分布是IB数学的核心主题，在分析与方法（AA）和应用与解释（AI）中都会出现。考官特别青睐考查我们能否构造概率分布表、计算期望值，以及运用二项分布和泊松分布等有名分布。本文按题型整理相关技能，从基础定义到依赖计算器的复杂问题，帮助你系统复习。

1. Random Variables and Discrete Distributions | 随机变量与离散分布

A random variable assigns a numerical value to each outcome of a random experiment. It is discrete if its possible values form a finite or countable set, such as 0, 1, 2, 3. Common examples include the number of heads in three coin tosses or the number of customers arriving in a minute.

随机变量将随机试验的每个结果对应为一个数值。若它可能的取值是有限个或可数无限个，则称为离散随机变量，例如抛三次硬币出现正面的次数，或一分钟内到达的顾客数。

• Key idea: write down all possible values of X before doing any probability calculation.

  关键思路：在做任何概率计算之前，先列出X的所有可能取值。

• Use capital X for the random variable and lower-case x for a specific value.

  用大写X表示随机变量，用小写x表示具体取值。

---

2. Conditions for a Valid Discrete Probability Distribution | 概率分布的合法性条件

A probability distribution is valid only if two conditions are satisfied. First, every individual probability must be between 0 and 1 inclusive. Second, the sum of all probabilities must equal exactly 1.

一个概率分布成立必须满足两个条件：第一，每个单独的概率都必须在0到1之间（含端点）；第二，所有概率之和必须恰好等于1。

0 ≤ P(X = x) ≤ 1 and Σ P(X = x) = 1

Typical exam question: a table is given with one missing value, and you must find it. Set up the equation: the sum of all known probabilities plus the missing value equals 1.

典型考题：给出一个分布表，其中有一个缺失值，要求你求它。只需要建立方程：所有已知概率之和加上缺失值等于1。

---

3. Representing Distributions: Tables, Graphs and Function Notation | 分布的表示：表格、图象与记号

Discrete distributions are usually displayed using a probability distribution table with rows for x and P(X = x). They can also be represented by a vertical line graph, where the height of each line shows the probability.

离散分布通常用概率分布表展示，表中包含x行和P(X = x)行。也可以用垂直线图表示，每条线的高度代表对应概率。

• Check that the graph or table covers every value of X.

  检查图象或表格是否覆盖X的所有取值。

• Use notation such as P(X = 2) or P(X ≤ 3) correctly.

  正确使用P(X = 2)或P(X ≤ 3)这样的记号。

• On the GDC, switch between table view and graph view to compare with your written answer.

  在计算器上，切换表格视图和图象视图，并与书面答案对照。

---

4. Expected Value μ and Variance σ² | 期望与方差

The expected value of a discrete random variable is a weighted average of all possible values, where the weights are the probabilities. It represents the long-run mean if the experiment is repeated many times.

离散随机变量的期望是全部可能取值的加权平均数，权重就是概率。它表示在大量重复试验下随机变量的长期平均水平。

E(X) = μ = Σ x · P(X = x)

Variance measures how spread out the distribution is. It can be found using the formula below. The standard deviation is the square root of variance.

方差衡量分布的离散程度，可用下面的公式计算。标准差是方差的平方根。

Var(X) = σ² = Σ (x − μ)² · P(X = x) = E(X²) − μ²

For E(X²), use the formula E(X²) = Σ x² · P(X = x). Many students forget to square the value x before multiplying by the probability.

计算E(X²)时使用公式E(X²) = Σ x² · P(X = x)。很多同学忘记先平方x再乘概率。

---

5. Binomial Distribution | 二项分布

The binomial distribution models the number of successes in a fixed number of independent trials. We write X ~ B(n, p), where n is the number of trials and p is the probability of success on each trial.

二项分布用来描述固定次数独立试验中成功次数的分布。记作X ~ B(n, p)，其中n是试验次数，p是单次试验的成功概率。

P(X = x) = C(n, x) · pˣ · (1 − p)ⁿ⁻ˣ

Four conditions must hold: there are n identical trials; each trial has two outcomes called success and failure; the trials are independent; p remains constant.

应用二项分布需满足四个条件：n次相同试验；每次试验只有成功与失败两种结果；各次试验相互独立；每次试验的成功概率p保持不变。

• Question type: “At least three” means P(X ≥ 3) = 1 − P(X ≤ 2).

  常见题型：“至少三次”即P(X ≥ 3) = 1 − P(X ≤ 2)。

• Question type: “Exactly two” means P(X = 2), found directly from the formula or GDC.

  常见题型：“恰好两次”即P(X = 2)，可直接使用公式或计算器。

---

6. Geometric Distribution | 几何分布

The geometric distribution counts the number of trials needed to achieve the first success. It is often written X ~ Geo(p). Unlike the binomial distribution, there is no fixed n.

几何分布表示首次成功所需试验的次数，常记作X ~ Geo(p)。与二项分布不同，它没有固定的试验总次数n。

P(X = x) = (1 − p)ˣ⁻¹ · p for x = 1, 2, 3, …

In this context, x is the trial number of the first success. For example, if p = 0.2 and you want the probability that the first success occurs on the 4th trial, set x = 4.

这里的x是首次成功出现的试验序号。例如p = 0.2，求第一次成功出现在第4次试验的概率，就令x = 4。

---

7. Poisson Distribution | 泊松分布

The Poisson distribution models the number of events occurring in a fixed interval of time or space, provided the events happen independently and at a constant average rate. We write X ~ Po(λ), where λ is the mean number of events.

泊松分布用于描述在固定时间或空间区间内事件发生的次数，前提是事件独立发生且平均发生率保持不变。记作X ~ Po(λ)，其中λ是事件发生的平均次数。

P(X = x) = e⁻ᶫ · λˣ / x!

Important: x! means x factorial. The parameter λ is both the mean and the variance of a Poisson distribution. This is a quick way to check if a Poisson model might be appropriate.

注意：x!表示x的阶乘。参数λ既是泊松分布的均值也是其方差。这个特点可用来快速判断泊松模型是否适用。

---

8. Mean and Variance of Named Distributions | 常用分布的均值与方差

Using the general E(X) and Var(X) definitions every time is slow. For named distributions, memorise the shortcut formulas below.

每次都用E(X)和Var(X)的定义式会很慢。对于有名分布，应当记住下面的快捷公式。

Distribution	E(X)	Var(X)
X ~ B(n, p)	np	np(1 − p)
X ~ Geo(p)	1 / p	(1 − p) / p²
X ~ Po(λ)	λ	λ

For binomial and Poisson, the formulas for mean and variance are often tested directly. For geometric distribution, only the mean appears in some syllabuses, so check your course specification.

二项分布和泊松分布的均值与方差公式经常直接考查。几何分布在某些考纲中只要求均值，请务必查看你所学科目的教学大纲。

---

9. GDC Skills and When to Use Them | 计算器使用技巧

Modern IB papers allow a GDC for many probability questions. However, you must still show enough written reasoning to justify your answer. Learn the menu paths for binomial, Poisson and geometric probability commands.

现代IB考试中，许多概率题可以使用图形计算器，但仍然需要写出足够的过程来支持答案。请熟悉计算器中二项分布、泊松分布和几何分布概率命令的菜单路径。

• Binomial PDF: computes P(X = k) for B(n, p).

  二项概率密度函数：计算B(n, p)中的P(X = k)。

• Binomial CDF: computes P(X ≤ k), then use complement for P(X > k).

  二项累积分布函数：计算P(X ≤ k)，求P(X > k)时使用补事件。

• Poisson CDF: computes P(X ≤ k), enabling questions like P(3 ≤ X ≤ 7).

  泊松累积分布函数：计算P(X ≤ k)，可用于求P(3 ≤ X ≤ 7)这类问题。

• Always write the distribution you used, for example X ~ B(10, 0.3), before the GDC result.

  在写出计算器结果之前，先写出所用分布，例如X ~ B(10, 0.3)。

---

10. Common Exam Traps | 常见考试陷阱

One frequent trap is confusing P(X = k) with P(X ≤ k). Another is forgetting to convert “at most” and “at least” into the correct inequality. Also, when a question says “find the expected value of 2X + 3”, use linearity rules rather than recalculating from scratch.

一个常见陷阱是混淆P(X = k)与P(X ≤ k)。另一个是忘记将“至多”“至少”转换成正确的不等式。此外，当题目要求“求2X + 3的期望”时，应使用线性运算规则，而不是从头重新计算。

• E(aX + b) = aE(X) + b

  E(aX + b) = aE(X) + b

• Var(aX + b) = a²Var(X), because adding a constant does not change spread.

  Var(aX + b) = a²Var(X)，因为加上常数不会改变离散程度。

• Do not use binomial distribution when trials are not independent or when p changes.

  当试验不独立或概率p变化时，不要使用二项分布。

---

11. Worked Example: Full Marks Solution | 完整例题示范

A fair six-sided die is rolled 12 times. Let X be the number of times a 5 is obtained. Find P(X = 3) and P(X ≥ 2).

一枚均匀六面骰子掷12次。令X表示出现5点的次数。求P(X = 3)和P(X ≥ 2)。

Step 1: Identify the distribution. Each roll is independent, there are two outcomes of interest, n = 12 and p = 1/6. So X ~ B(12, 1/6).

第一步：确定分布。每次掷骰独立，关心的结果只有“出现5”和“不出现5”，n = 12，p = 1/6。因此X ~ B(12, 1/6)。

P(X = 3) = C(12, 3) · (1/6)³ · (5/6)⁹ ≈ 0.1974

Step 2: For P(X ≥ 2), use the complement. P(X ≥ 2) = 1 − P(X = 0) − P(X = 1).

第二步：求P(X ≥ 2)时使用补事件。P(X ≥ 2) = 1 − P(X = 0) − P(X = 1)。

P(X ≥ 2) = 1 − (5/6)¹² − 12 · (1/6) · (5/6)¹¹ ≈ 0.6187

Always check that your final probability is between 0 and 1 and that you have quoted the distribution clearly.

始终检查最终概率在0到1之间，并清楚写出所用分布。

---

12. Study Strategy for Exam Success | 备考策略

Discrete probability distribution questions reward organised working. Start by defining X, then list possible values, then state the distribution model and parameters, then calculate, then interpret the result.

离散概率分布题重视有条理的书写。建议先定义X，再列出可能取值，再说明分布模型和参数，然后计算，最后解释结果。

• Practise reading inequality statements: “fewer than”, “more than”, “at most”, “at least”.

  练习读不等式表达：“少于”“多于”“至多”“至少”。

• Memorise all formula sheet and non-formula-sheet formulas before the exam.

  考试前熟记公式表内和公式表外需要记忆的公式。

• Use past paper questions to identify your school’s most common question style.

  通过历年真题找出你的学校最常考的题型风格。

• Rewrite answer lines until they are clear enough for another student to follow.

  反复改写解答过程，直到另一位同学也能看懂。

Mastering discrete probability distributions is not just about memorising formulas. The real skill is translating a real-world situation into the correct distribution and the correct probability statement. Once you can do that, most IB exam questions become routine.

掌握离散概率分布不只是记住公式。真正的技能在于把实际问题转化为正确的分布和正确的概率表达式。一旦你做到这一点，大部分IB考题都会变得非常常规。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）