The Binomial Distribution | 二项分布

📚 The Binomial Distribution | 二项分布

The binomial distribution is one of the most important discrete probability models in the Edexcel A Level Mathematics specification. It counts the number of successes in a fixed number of independent trials, where each trial has the same probability of success. This article covers its conditions, probability formula, mean and variance, cumulative probabilities, and common exam techniques.

二项分布是爱德思 A Level 数学考试大纲中最重要的离散概率模型之一。它统计在固定次数的独立试验中成功的次数，并且每次试验的成功概率相同。本文将介绍它的使用条件、概率公式、均值与方差、累积概率以及常见考试技巧。

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

The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials. ‘Success’ does not have to mean a good outcome; it is simply the outcome we are counting.

二项分布是一种离散概率分布，描述在固定次数的独立试验中成功的次数。“成功”不一定表示好的结果，它只是我们要计数的那个结果。

For example, if you toss a fair coin 10 times and count the number of heads, the total number of heads follows a binomial distribution.

例如，如果你抛一枚均匀硬币 10 次并记录正面朝上的次数，正面次数就服从二项分布。

Other real-life situations include the number of defective items in a batch, the number of patients who recover after a treatment, or the number of sixes rolled in 20 dice throws.

其他实际例子包括一批产品中次品的数量、接受治疗后康复的患者人数，或者掷骰子 20 次出现 6 点的次数。

2. Conditions for a Binomial Model | 二项模型的条件

A situation can be modelled by a binomial distribution only if all four conditions hold:

只有同时满足以下四个条件，才能用二项分布建模：

1. There is a fixed number of trials, n.

1. 试验次数 n 是固定的。

2. Each trial has exactly two outcomes, usually called success and failure.

2.

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