Binomial Distribution | 二项分布

📚 Binomial Distribution | 二项分布

The binomial distribution is a cornerstone of discrete probability theory. It describes the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: success (with probability p) or failure (with probability q = 1 − p).

二项分布是离散概率论的基石之一。它描述了在 n 次独立试验中成功的次数，每次试验都问一个是非问题，每次结果要么成功（概率 p），要么失败（概率 q = 1 − p）。

In IB Mathematics, students encounter the binomial distribution in both the Analysis & Approaches (AA) and Applications & Interpretation (AI) courses. It forms the basis for hypothesis testing and connects to the normal distribution through approximations.

在 IB 数学中，无论是分析与方法 (AA) 还是应用与解释 (AI) 课程，学生都会接触到二项分布。它是假设检验的基础，并通过近似与正态分布建立联系。

1. What is a Binomial Distribution? | 什么是二项分布？

A random variable X follows a binomial distribution if it counts the number of successes in n independent Bernoulli trials, each with the same probability of success p. The possible values of X are 0, 1, 2, …, n.

如果随机变量 X 统计的是 n 次独立伯努利试验中成功的次数，且每次成功的概率 p 相同，则 X 服从二项分布。X 的可能取值为 0, 1, 2, …, n。

We write X ~ B(n, p) to indicate that X has a binomial distribution with parameters n and p. The number n is often called the number of trials, and p is the probability of success on each trial.

我们记作 X ~ B(n, p) 表示 X 服从参数为 n 和 p 的二项分布。

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