📚 Geometric Distribution | 几何分布
The geometric distribution models the number of trials needed to achieve the first success in a sequence of independent Bernoulli trials with constant success probability p. It is one of the fundamental discrete probability distributions studied in IB Mathematics, particularly in the Analysis and Approaches HL and Applications and Interpretation HL courses. This article provides a complete revision guide covering the definition, key formulas, properties, and exam-style problem-solving strategies.
几何分布描述的是在一系列独立伯努利试验中，首次成功所需的试验次数，其中每次试验的成功概率 p 保持不变。它是 IB 数学（特别是 AA HL 和 AI HL 课程）中重要的离散概率分布之一。本文将提供完整的复习指南，涵盖定义、核心公式、性质以及考试风格的解题策略。
1. Definition and Notation | 定义与记号
1 − p). Let the random variable X denote the number of trials up to and including the first success. Then X follows a geometric distribution with parameter p, written as X ~ Geo(p). The possible values of X are the positive integers 1, 2, 3, …
假设我们独立地重复一个试验，每次试验只有两种结果：成功（概率为 p）或失败（概率为 q = 1 − p）。设随机变量 X 表示直到首次成功（包括该次）为止的试验次数。那么 X 服从参数为 p 的几何分布，记作 X ~ Geo(p)。X 的可能取值为正整数 1, 2, 3, …
A different convention defines the geometric distribution as the number of failures before the first success, with support {0, 1, 2, …}. In IB Mathematics, the first convention (number of trials) is standard; however, you should always read the question carefully to determine which definition is being used.
另一种约定将几何分布定义为首次成功之前失败的次数，其取值范围为 {0, 1, 2, …}。在 IB 数学中，第一种约定（试验次数）是标准定义；但做题时务必仔细阅读题目，确认真实使用哪种定义。
2. Probability Mass Function | 概率质量函数
For a random variable X ~ Geo(p), the probability that the first success occurs on the r-th trial is given by the probability mass function (PMF):
设随机变量 X ~ Geo(p)，则首次成功发生在第 r 次试验的概率由如下概率质量函数（PMF）给出：
P(X = r) = (1 − p)r−1p, for r = 1, 2, 3, …
This formula is derived by multiplying the probabilities of r − 1 consecutive failures, each with probability (1 − p), followed by one success with probability p. Since all trials are independent, the multiplication rule applies directly.
该公式由连续 r − 1 次失败（每次概率为 1 − p）之后一次成功（概率为 p）的概率相乘得出。由于每次试验相互独立，可直接应用乘法法则。
To verify that this is a valid probability distribution, we sum the probabilities over all possible values of r. Using the infinite geometric series formula:
为了验证这是一个合法的概率分布，我们将概率对所有可能的 r 求和。利用无穷等比数列求和公式：
Σr=1∞ p(1 − p)r−1 = p × 1/[1 − (1 − p)] = 1
The distribution gets its name from the geometric series used in this summation.
该分布的名称正来源于上述求和中所使用的等比级数。
3. Conditions for Using the Geometric Distribution | 几何分布的适用条件
To apply the geometric distribution, the following conditions must be satisfied:
应用几何分布需要满足以下条件：
Each trial has exactly two possible outcomes: success or failure.
每次试验只有两种可能结果：成功或失败。
The probability of success p is the same for every trial.
每次试验的成功概率 p 保持不变。
The trials are independent — the outcome of one trial does not affect any other.
各次试验相互独立——一次试验的结果不影响其他任何一次。
The experiment continues until the first success occurs (there is no fixed number of trials).
试验一直重复到首次成功出现为止（没有固定的试验总次数）。
These conditions are identical to those of the binomial distribution, except that the binomial distribution fixes the total number of trials n, whereas the geometric distribution does not.
这些条件与二项分布的条件完全相同，区别在于二项分布固定了试验总次数 n，而几何分布没有固定试验次数。
4. Cumulative Distribution Function | 累积分布函数
The cumulative distribution function (CDF) gives the probability that the first success occurs by the r-th trial. This is equal to the probability that at least one success has occurred in the first r trials:
累积分布函数（CDF）给出首次成功出现在第 r 次或之前（即前 r 次内）的概率，等于前 r 次试验中至少出现一次成功的概率：
P(X ≤ r) = 1 − P(no success in first r trials) = 1 − (1 − p)r
Equivalently, the complement represents the probability that the first success occurs after the r-th trial:
等价地，其补事件表示首次成功发生在第 r 次之后的概率：
P(X > r) = (1 − p)r
as are extremely useful in exam problems. For example, to find P(3 ≤ X ≤ 6), we can write:
这两个公式在考试中非常有用。例如，要求 P(3 ≤ X ≤ 6)，我们可以写成：
P(3 ≤ X ≤ 6) = P(X ≤ 6) − P(X ≤ 2) = [1 − (1 − p)6] − [1 − (1 − p)2]
, not P(X ≤ 3), because the event X ≤ 3 includes X = 3, which should be kept in the range.
注意：要减去的是 P(X ≤ 2)，而不是 P(X ≤ 3)，因为事件 X ≤ 3 包含 X = 3，而 X = 3 应保留在所求范围内。
5. Expected Value | 期望值
The mean (expected value) of a geometric distribution is the expected number of trials until the first success. Intuitively, if the probability of success is p, the expected waiting time is the reciprocal of p:
几何分布的均值（期望值）是直到首次成功为止的期望试验次数。直观地说，如果成功的概率为 p，则期望等待时间为 p 的倒数：
E(X) = 1/p
This result can be derived from the definition of expectation:
该结果可由期望的定义推导：
E(X) = Σr=1∞ r · p(1 − p)r−1 = 1/p
s the most likely value; it is simply the long-run average number of trials needed across many repetitions.
例如，反复掷一颗公平的六面骰子直到出现 6，p = 1/6，因此期望投掷次数为 E(X) = 1/(1/6) = 6。这并不意味着 6 是最可能出现的具体次数；它只是长期重复试验所需次数的平均值。
6. Variance | 方差
The variance measures the spread of the distribution, that is, how far the number of trials is typically from the mean. For X ~ Geo(p):
方差衡量分布的离散程度，即试验次数一般偏离均值多远。对于 X ~ Geo(p)：
Var(X) = (1 − p) / p²
Consequently, the standard deviation is:
因此，标准差为：
SD(X) = √1 − p / p
Notice that the variance is large when p is small, because a small success probability means the number of trials can be highly variable — some runs may succeed very quickly, while others may require many attempts.
注意：当 p 较小时方差很大，因为较小的成功概率意味着试验次数可能有很大波动——有些试验序列很快成功，而另一些则需要大量尝试。
= 20, so SD(X) ≈ 4.47. The high spread reflects the unpredictability of the waiting time.
例如，取一枚偏置硬币，p = 0.2。则 E(X) = 5，Var(X) = 0.8 / 0.04 = 20，因此 SD(X) ≈ 4.47。较大的离散程度反映了等待时间的不可预测性。
7. The Memoryless Property | 无记忆性
A hallmark of the geometric distribution is its memoryless property. If the first success has not occurred in the first m trials, the probability that we still need more than n additional trials is the same as the probability that the original waiting time was more than n:
几何分布的一个标志性特征是其无记忆性。如果前 m 次试验中尚未出现首次成功，那么接下来仍需要超过 n 次试验的概率，与初始等待时间超过 n 的概率相同：
P(X > m + n | X > m) = P(X > n) = (1 − p)n
In words, after several consecutive failures, "forgetting" the past failures does not change the future waiting probability. The process restarts fresh each time because every trial is independent.
用语言来表达：连续经历若干次失败之后，可以“忘记”过去的失败，因为未来等待的概率并不改变。由于每次试验都是独立的，过程每次都重新开始。
To see why this is true, use the conditional probability formula:
为理解这一点为何成立，使用条件概率公式：
P(X > m+n and X > m) / P(X > m) = (1 − p)m+n / (1 − p)m = (1 − p)n
The memoryless property is unique to the geometric distribution among discrete distributions that count trials. It is frequently tested in IB exam questions involving conditional probability.
在统计试验次数的离散分布中，无记忆性是几何分布独有的性质。IB 考试经常在条件概率题中考查这一性质。
8. Relation to the Binomial Distribution | 与二项分布的关系
The geometric distribution and the binomial distribution both arise from independent Bernoulli trials, but they answer different questions:
几何分布与二项分布都源自独立的伯努利试验，但它们回答的是不同的问题：
The binomial distribution X ~ B(n, p) counts the number of successes in a fixed number of trials n.
二项分布 X ~ B(n, p) 统计固定次数 n 次试验中的成功次数。
The geometric distribution X ~ Geo(p) counts the number of trials until the first success, with no fixed upper limit.
几何分布 X ~ Geo(p) 统计直到首次成功所需的试验次数，没有固定上限。
There is also a deeper connection. If we conduct n independent trials and each has success probability p, then the probability of no success in the first r − 1 trials and success on trial r is exactly the geometric probability. Thus, the geometric distribution is the distribution of the "waiting time" for a binomial-type process.
两者还有更深层的联系。若进行 n 次独立试验，每次成功概率为 p，则前 r − 1 次无成功且第 r 次成功的概率正是几何分布的概率。因此，几何分布可看作二项型过程的“等待时间”分布。
Another useful relationship: for fixed n, P(X ≤ n) = 1 − (1 − p)n quals the binomial probability that at least one success occurs, that is, P(Y ≥ 1) where Y ~ B(n, p). This link often appears in exam questions that combine both distributions.
另一个有用的关系：对于固定的 n，P(X ≤ n) = 1 − (1 − p)n，恰好等于二项分布中至少出现一次成功的概率，即 Y ~ B(n, p) 时 P(Y ≥ 1)。这个联系经常出现在同时涉及两种分布的考试题中。
9. Worked Example | 例题分析
A basketball player has a free-throw success rate of 70%.
一名篮球运动员的罚球命中率为 70%。
(a) Find the probability that the player makes the first successful free throw on the 3rd attempt.
(a) 求该运动员在第 3 次尝试中才首次命中的概率。
Here X ~ Geo(0.7). Using the PMF with r = 3:
这里 X ~ Geo(0.7)。对 r = 3 使用概率质量函数：
P(X = 3) = (0.3)² × 0.7 = 0.063
(b) Find the probability that the first success occurs within the first 3 attempts.
(b) 求首次命中出现在前 3 次尝试中的概率。
P(X ≤ 3) = 1 − (0.3)³ = 1 − 0.027 = 0.973
(c) Find the expected number of attempts and the standard deviation.
(c) 求期望尝试次数和标准差。
E(X) = 1/0.7 ≈ 1.43, SD(X) = √0.3/0.7 ≈ 0.782
(d) Given that the player missed the first two shots, find the probability that she will not score until after the 5th attempt.
(d) 已知该运动员前两投未中，求她直到第 5 次尝试之后才首次命中的概率。
This is a direct application of the memoryless property. We need P(X > 5 | X > 2) = P(X > 3) = (0.3)³ = 0.027.
这是无记忆性的直接应用。需要求 P(X > 5 | X > 2) = P(X > 3) = (0.3)³ = 0.027。
This worked example covers the PMF, CDF, expectation, standard deviation, and the memoryless property — all essential exam skills.
该例题涵盖了概率质量函数、累积分布函数、期望、标准差和无记忆性——这些都是考试中的关键技能。
10. Common Mistakes and Exam Tips | 常见错误与考试提示
IB students frequently make the following errors when working with geometric distributions:
IB 学生在处理几何分布时常犯以下错误：
Confusing the two definitions. If X counts trials until first success, the PMF is p(1 − p)r−1. If X counts failures before the first success, the PMF is p(1 − p)r. Always check the stated support set.
混淆两种定义。 若 X 表示直到首次成功的试验次数，则 PMF 为 p(1 − p)r−1；若 X 表示首次成功前的失败次数，则 PMF 为 p(1 − p)r。务必检查题目给出的取值范围。
Forgetting the final success. In P(X = r), the r-th trial must be a success, so the exponent of (1 − p) is r − 1, not r.
忘记最后一次是成功。 在 P(X = r) 中，第 r 次试验必须是成功的，因此 (1 − p) 的指数是 r − 1，而不是 r。
Misapplying the CDF. The formula P(X ≤ r) = 1 − (1 − p)r counts successes from trial 1 to r; it is not the same as P(X = r).
误用累积分布函数。 公式 P(X ≤ r) = 1 − (1 − p)r 统计的是第 1 次到第 r 次出现成功的概率，它不等于 P(X = r)。
Rounding probabilities too early. In multi-part questions, carry exact fractions or keep at least 3 significant figures until the final answer.
过早四舍五入概率。 在多部分题目中，建议保留精确分数或至少保留 3 位有效数字，直到算出最终答案。
Using the normal distribution as an approximation. The geometric distribution is not approximated by the normal unless p is very small and n is large, and this is beyond the IB syllabus. Use exact formulas.
用正态分布近似。 几何分布通常不使用正态近似，除非 p 很小且 n 很大，而这不属于 IB 课程范围。应使用精确公式。
Exam tips: write down the distribution statement "X ~ Geo(p)" first, state the parameter clearly, and quote the formula you are using. Marks are often awarded for method and working, not just the final numerical answer.
考试提示：先写出分布声明“X ~ Geo(p)”，明确参数，并注明所使用的公式。评分往往看重方法和过程，而不仅仅是最终数值。
11. Summary | 总结
The geometric distribution is a fundamental discrete model in IB Mathematics. It counts the trials needed to obtain the first success in a sequence of independent Bernoulli trials. Its key formulas are:
几何分布是 IB 数学中一个基础的离散模型。它统计在独立伯努利试验系列中获得首次成功所需的试验次数。其核心公式如下：
| Property | Formula | 中文性质 | 公式 |
| PMF | P(X = r) = (1 − p)r−1p | 概率质量函数 | P(X = r) = (1 − p)r−1p |
| CDF | P(X ≤ r) = 1 − (1 − p)r | 累积分布函数 | P(X ≤ r) = 1 − (1 − p)r |
| Mean | E(X) = 1/p | 期望 | E(X) = 1/p |
| Variance | Var(X) = (1 − p)/p² | 方差 | Var(X) = (1 − p)/p² |
| Memoryless | P(X > m+n | X > m) = P(X > n) | 无记忆性 | P(X > m+n | X > m) = P(X > n) |
Mastering the distinction between the geometric and binomial distributions, applying the memoryless property with confidence, and practising CDF calculations will help you earn full marks in this topic. Always set up your distribution clearly and show every step of your working.
掌握几何分布与二项分布的区别、自信地运用无记忆性，并勤加练习累积分布函数计算，将帮助你在该知识点上获得满分。做题时务必清晰设定分布，并展示每一步计算过程。
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