📚 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
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
(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, …
.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
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)。
on. 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。
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