📚 F – The Binomial Distribution | F – 二项分布
The binomial distribution is a fundamental discrete probability distribution that describes the number of successes in a fixed number of independent trials, each with the same probability of success. It appears extensively in both IB Mathematics: Analysis and Approaches (AA) and Applications and Interpretation (AI), forming the basis for statistical inference and hypothesis testing.
二项分布是一种基本的离散概率分布，它描述了在固定次数的独立试验中，每次试验具有相同成功概率时成功的次数。二项分布在 IB 数学：分析与方法 (AA) 和应用与解释 (AI) 中均有广泛涉及，是统计推断和假设检验的基础。
1. Bernoulli Trials and the Binomial Setting | 伯努利试验与二项情境
A single random experiment with exactly two possible outcomes – success and failure – 1 − p. Repeating such trials independently a fixed number of times, n, and counting the total number of successes leads to a binomial distribution.
只有两个可能结果（成功和失败）的单次随机实验称为伯努利试验。如果用 p 表示成功的概率，那么失败的概率就是 q = 1 − p。将这样的试验独立地重复固定的次数 n，并统计成功的总次数，就得到二项分布。
We write X ~ B(n, p) to indicate that the random variable X follows a binomial distribution with parameters n and p. Here n is the number of trials, and p is the constant probability of success in each trial.
我们记作 X ~ B(n, p)，表示随机变量 X 服从参数为 n 和 p 的二项分布。其中 n 是试验次数，p 是每次试验成功的恒定概率。
2. Conditions for a Binomial Experiment | 二项试验的条件
1. There is a fixed number, n, of identical trials. The number of trials is known in advance and does not change during the experiment.
1. 存在一个固定的试验次数 n。试验次数是事先已知的，且在实验过程中不会改变。
2. The trials are independent. The outcome of any trial does not influence the outcomes of other trials.
2. 各次试验相互独立。任何一次试验的结果不会影响其他试验的结果。
3. Each trial has only two possible outcomes, conventionally called success and failure.
3. 每次试验只有两种可能的结果，通常称为成功和失败。
4. The probability of success, p, remains constant across all trials. This condition is essential for the binomial model to be exact.
4. 成功的概率 p 在所有试验中保持恒定。这一条件是二项模型准确成立的关键。
3. Notation and Parameters | 符号与参数
The two parameters that fully define a binomial distribution are n (the number of trials) and p (the probability of success on a single trial). Once these are known, the entire probability distribution of the number of successes, X, can be determined.
完全定义一个二项分布的两个参数是 n（试验次数）和 p（单次试验成功的概率）。一旦知晓这两个参数，成功次数 X 的整个概率分布就可以确定。
It is common to denote the binomial random variable as X ~ B(n, p). For example, X ~ B(10, 0.3) means that we repeat a Bernoulli trial 10 times, where each trial has a 0.3 chance of success, and X counts the number of successes.
通常将二项随机变量记为 X ~ B(n, p)。例如，X ~ B(10, 0.3) 表示我们将一个伯努利试验重复 10 次，每次试验成功的概率为 0.3，X 统计成功的次数。
4. Binomial Probability Formula | 二项概率公式
For a binomial random variable X ~ B(n, p), the probability of obtaining exactly k successes (where k = 0, 1, 2, …, n) is given by the probability mass function:
对于二项随机变量 X ~ B(n, p)，恰好获得 k 次成功的概率（k = 0, 1, 2, …, n）由概率质量函数给出：
P(X = k) = ⁿCₖ × pᵏ × (1 − p)ⁿ⁻ᵏ
The coefficient ⁿCₖ (read as 'n choose k' the number of ways to choose k successes out of n trials and can be calculated as ⁿCₖ = n! / (k! × (n − k)! and (1 − p)ⁿ⁻ᵏ is the probability of the remaining n − k trials all resulting in failure.
系数 ⁿCₖ（读作 “n 选 k”）是从 n 次试验中选出 k 次成功的方式数，计算公式为 ⁿCₖ = n! / (k! × (n − k)!)。项 pᵏ 是这 k 次成功全部发生的概率，而 (1 − p)ⁿ⁻ᵏ 是其余 n − k 次试验均失败的概率。
5. Worked Example – Exact Probability | 计算实例 – 精确概率
ess is defined as rolling a six, so p = 1/6. Let X be the number of sixes obtained. Then X ~ B(8, 1/6). Calculate the probability of rolling exactly three sixes.
一个均匀的骰子被投掷 8 次。成功定义为掷出六点，因此 p = 1/6。设 X 为得到的六点的次数，则 X ~ B(8, 1/6)。计算恰好掷出三个六点的概率。
= 56 × (1/216) × 0.4019 ≈ 0.1042. So the probability of exactly three sixes is about 10.4%.
使用公式：P(X = 3) = ⁸C₃ × (1/6)³ × (5/6)⁵。首先计算 ⁸C₃ = 56，然后 (1/6)³ = 1/216，(5/6)⁵ ≈ 0.4019。相乘得 P(X = 3) = 56 × (1/216) × 0.4019 ≈ 0.1042。因此恰好得到三个六点的概率约为 10.4%。
The probability of getting exactly 5 correct is P(X = 5) = ¹²C₅ × 0.25⁵ × 0.75⁷ ≈ 0.1032.
再举一例，假设一份选择题测验有 12 题，每题 4 个选项。一名学生完全随机猜测。那么 X = 正确题数服从 B(12, 0.25)。恰好答对 5 题的概率为 P(X = 5) = ¹²C₅ × 0.25⁵ × 0.75⁷ ≈ 0.1032。
6. Expectation and Variance | 期望与方差
s intuitive result tells us the long-run average number of successes if the experiment were repeated many times.
二项分布中成功次数的期望值就是试验次数与成功概率的乘积：E(X) = n p。这一直观的结果告诉我们，如果实验重复很多次，长期成功的平均次数就是 np。
(1 − p)). Note that the variance is largest when p = 0.5, reflecting maximum uncertainty.
方差衡量分布的离散程度，Var(X) = n p (1 − p)。因此标准差为 σ = √(n p (1 − p))。注意当 p = 0.5 时方差最大，这反映了最大的不确定性。
For the dice example with X ~ n average, we expect about 1.33 sixes in 8 rolls, with a standard deviation of about 1.05.
在骰子例子 X ~ B(8, 1/6) 中，E(X) = 8 × 1/6 ≈ 1.333，Var(X) = 8 × (1/6) × (5/6) ≈ 1.111，σ ≈ 1.054。平均而言，我们期望在 8 次投掷中得到约 1.33 个六点，标准差约为 1.05。
7. Shape of the Binomial Distribution | 二项分布的形状
tribution is perfectly symmetrical. When p < 0.5, the distribution is positively skewed (skewed to the right), with a tail extending towards larger values of X. When p > 0.5, the distribution is negatively skewed (skewed to the left).
二项分布的形状取决于 n 和 p。当 p = 0.5 时，分布是完全对称的。当 p < 0.5 时，分布呈正偏态（右偏），尾部向较大的 X 值延伸。当 p > 0.5 时，分布呈负偏态（左偏）。
As the number of trials n increases, the binomial distribution becomes more bell-shaped and can be approximated by a normal distribution under certain conditions (see Section 10). visualising the shape helps in understanding the probabilities of extreme values.
随着试验次数 n 的增加，二项分布变得更加接近钟形，并且在特定条件下可以用正态分布来近似（见第 10 节）。观察分布的形状有助于理解极端值出现的概率。
8. Using a GDC for Binomial Calculations | 使用图形计算器
In IB examinations, you are expected to use a graphical display calculator (GDC) for most binomial probability questions. The key functions are binompdf and binomcdf.
在 IB 考试中，你需要使用图形计算器 (GDC) 来解决大多数二项概率问题。关键的函数是 binompdf 和 binomcdf。
binompdf(n, p, k) function calculates the exact probability P(X = k). For instance, for X ~ B(12, 0.25), binompdf(12, 0.25, 5) returns the value 0.1032 for exactly 5 successes.
函数 binompdf(n, p, k) 计算精确概率 P(X = k)。例如，对于 X ~ B(12, 0.25)，binompdf(12, 0.25, 5) 会返回恰好 5 次成功的概率值 0.1032。
f(12, 0.25, 5) gives P(X ≤ 5). If you need P(X < 5), use P(X ≤ 4) = binomcdf(12, 0.25, 4).
函数 binomcdf(n, p, k) 计算累积概率 P(X ≤ k)，即最多 k 次成功的概率。例如，binomcdf(12, 0.25, 5) 给出 P(X ≤ 5)。如果需要 P(X < 5)，则使用 P(X ≤ 4) = binomcdf(12, 0.25, 4)。
9. Cumulative Probabilities and Inequalities | 累积概率与不等式
Many IB-style questions ask for probabilities involving inequalities. The key relationships using binomcdf are:
许多 IB 试题会要求计算涉及不等式的概率。使用 binomcdf 的关键关系如下：
≥ k) is found by the complement: P(X ≥ k) = 1 − P(X ≤ k − 1) = 1 − binomcdf(n, p, k − 1).
P(X ≤ k) = binomcdf(n, p, k) 可直接从计算器得到。P(X ≥ k) 通过补集求得：P(X ≥ k) = 1 − P(X ≤ k − 1) = 1 − binomcdf(n, p, k − 1)。
t since X is discrete, P(X < k) = P(X ≤ k − 1) and P(X > k) = P(X ≥ k + 1) = 1 − P(X ≤ k).
对于严格不等式，要注意 X 是离散的，因此 P(X < k) = P(X ≤ k − 1)，而 P(X > k) = P(X ≥ k + 1) = 1 − P(X ≤ k)。
A typical question: If X ~ 6 and 10 inclusive. This is P(6 ≤ X ≤ 10) = binomcdf(20, 0.4, 10) − binomcdf(20, 0.4, 5).
一个典型的问题：若 X ~ B(20, 0.4)，求 X 在 6 到 10 之间（含两端）的概率。即 P(6 ≤ X ≤ 10) = binomcdf(20, 0.4, 10) − binomcdf(20, 0.4, 5)。
10. Normal Approximation to the Binomial | 二项分布的正态近似
trials n is large enough so that both np ≥ 5 and n(1 − p) ≥ 5, the binomial distribution X ~ variance σ² = np(1 − p). This is useful for avoiding lengthy calculations when n is large.
当试验次数 n 足够大，满足 np ≥ 5 且 n(1 − p) ≥ 5 时，二项分布 X ~ B(n, p) 可以用均值为 μ = np、方差为 σ² = np(1 − p) 的正态分布来近似。这在 n 较大时可避免冗长的计算。
Because the binomial is discrete and the normal is continuous, a continuity correction must be
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