📚 IB Math: Geometric Distribution Model and Typical Examples | IB数学：几何分布模型与典型例题
The geometric distribution is a core discrete probability distribution in IB Mathematics, appearing in both Analysis and Approaches (AA) and Applications and Interpretation (AI). It models the number of independent Bernoulli trials required to achieve the first success. This article provides a comprehensive review of the model, its key formulas, common pitfalls, and step-by-step worked examples tailored for IB exam preparation.
几何分布是IB数学中一个核心的离散概率分布，在分析与方法（AA）以及应用与解释（AI）中均有涉及。它用于描述在独立重复的伯努利试验中，获得首次成功所需的试验次数。本文将全面梳理该模型的核心公式、常见误区，并结合IB考试风格提供详细的典型例题解析。
1. What is a Geometric Distribution? | 什么是几何分布？
A geometric distribution is a discrete probability distribution that calculates the probability of the number of trials needed to get the first success in a sequence of independent and identically distributed Bernoulli trials. Each trial has exactly two outcomes: success or failure.
几何分布是一种离散概率分布，用于计算在一系列独立同分布的伯努利试验中，首次成功发生所需的试验次数的概率。每次试验只有两种结果：成功或失败。
For example, repeatedly tossing a fair coin until a head appears. The number of tosses needed follows a geometric distribution. If you get a head on the third toss, then the number of trials is 3.
例如，反复抛掷一枚均匀硬币直到出现正面。所需抛掷次数服从几何分布。如果你在第3次抛掷时才得到正面，那么试验次数就是3。
In IB, the random variable X is usually defined as the number of trials up to and including the first success. Thus, X can take values 1, 2, 3, and so on, without an upper limit.
在IB课程中，随机变量 X 通常被定义为“直到首次成功（包含该次）为止的总试验次数”。因此，X 可以取 1, 2, 3 等值，且没有上限。
2. Parameters and Conditions | 参数与适用条件
The geometric distribution has a single parameter, p, which represents the probability of success on each individual trial. For a valid geometric model, the following conditions must hold:
几何分布只有一个参数 p，它表示每一次试验中成功的概率。要使用几何分布模型，必须满足以下条件：
- Each trial has only two possible outcomes: success or failure.
- 每次试验只有两种可能的结果：成功或失败。
- The probability of success, p, is constant from trial to trial.
- 每次试验成功的概率 p 保持恒定不变。
- The trials are independent, meaning the outcome of one trial does not affect any other trial.
- 各次试验相互独立，即某一次试验的结果不会影响其他任何一次试验。
These conditions are identical to the requirements for a binomial distribution. The key difference lies in the question being asked: binomial counts successes in a fixed number of trials, while geometric counts trials until the first success.
这些条件与二项分布的要求完全相同。两者的核心区别在于所研究的问题：二项分布是在固定试验次数下统计成功次数；而几何分布则是统计直到首次成功出现时所需的试验次数。
3. The Probability Mass Function (PMF) | 概率质量函数（PMF）
If X follows a geometric distribution with parameter p, the probability that the first success occurs on the r-th trial is given by:
设 X 服从参数为 p 的几何分布，则首次成功发生在第 r 次试验的概率为：
P(X = r) = (1-p)ʳ⁻¹ p, for r = 1, 2, 3, …
This formula reflects a simple logic: the first r-1 trials must all be failures, each with probability (1-p), and then the r-th trial must be a success, with probability p. Since the trials are independent, we multiply these probabilities together.
这个公式背后的逻辑非常直观：前 r-1 次试验必须都是失败，每次失败的概率为 (1-p)；随后第 r 次试验必须成功，概率为 p。由于各次试验相互独立，我们将这些概率相乘即可。
For example, if p = 0.2, then P(X = 3) = (0.8)² × (0.2) = 0.128. This means there is a 12.8% chance that the first success occurs precisely on the third trial.
例如，若 p = 0.2，则 P(X = 3) = (0.8)² × (0.2) = 0.128。这意味着首次成功恰好发生在第三次试验的概率为12.8%。
4. Expectation and Variance | 期望与方差
The mean (expected value) and variance of a geometric distribution provide measures of the center and spread of the distribution. These are essential for solving IB exam questions that involve the long-run average behavior.
几何分布的期望（均值）和方差分别描述了分布的中心位置和离散程度。它们是解答IB考试中涉及长期平均行为题目的关键工具。
E(X) = 1/p
Var(X) = (1-p) / p²
For instance, if the probability of a student passing a driving test is 0.25 on any given attempt, the expected number of attempts needed to pass is E(X) = 1 / 0.25 = 4 attempts.
例如，若某学生每次参加驾照考试通过的概率为0.25，那么他通过考试所需尝试次数的期望值为 E(X) = 1 / 0.25 = 4 次。
The variance can be used to understand the variability. In this example, Var(X) = 0.75 / 0.0625 = 12, so the standard deviation is √12 ≈ 3.46 attempts.
方差帮助我们理解波动性。在该例中，Var(X) = 0.75 / 0.0625 = 12，因此标准差为 √12 ≈ 3.46 次。
5. The Memoryless Property | 无记忆性
A unique feature of the geometric distribution is the memoryless property. This means that the probability of needing additional trials is independent of how many failures have already occurred.
几何分布的一个独特性质是无记忆性。这意味着“还需要多少试验次数才能成功”的概率，与已经发生了多少次失败无关。
Formally, this is expressed as:
其数学表达形式为：
P(X > a + b | X > a) = P(X > b)
In terms of the geometric distribution, since P(X > k) = (1-p)ᵏ, the memoryless property is verified algebraically:
对于几何分布而言，因为 P(X > k) = (1-p)ᵏ，我们可以从代数角度验证无记忆性：
P(X > a + b) / P(X > a) = (1-p)^(a+b) / (1-p)ᵃ = (1-p)ᵇ = P(X > b)
This property is extremely useful in IB questions that ask for conditional probabilities, allowing us to “reset” the problem after a certain number of failures.
这个性质在IB考试的条件概率问题中非常实用，它允许我们在已知若干次失败后，将问题“重新归零”再进行计算。
6. Geometric Distribution vs. Binomial Distribution | 几何分布 vs 二项分布
Students often confuse geometric and binomial distributions because they both involve repeated Bernoulli trials. The table below highlights the key differences.
同学们经常混淆几何分布和二项分布，因为它们都涉及重复的伯努利试验。下表清晰地展示了它们的主要区别。
| 特征 Feature | 二项分布 B(n, p) | 几何分布 Geo(p) |
| 变量含义 Meaning | n 次试验中成功的次数 X | 首次成功所需的试验次数 X |
| 取值范围 Values | X = 0, 1, 2, …, n |
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