IB数学：二项分布公式与应用

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TITLE: IB Mathematics: Binomial Distribution Formulas and Applications | IB数学：二项分布公式与应用

📚 IB Mathematics: Binomial Distribution Formulas and Applications | IB数学：二项分布公式与应用

The binomial distribution is one of the most important discrete probability distributions in IB Mathematics. It models the number of successes in a fixed number of independent trials, each with the same probability of success. Understanding its formula, conditions, and applications is essential for success on both the Analysis and Approaches (AA) and Applications and Interpretation (AI) exams.

二项分布是IB数学中最重要的一类离散概率分布之一。它用于建立固定次数独立试验中成功次数的概率模型，且每次试验的成功概率相同。掌握其公式、适用条件与实际应用，对于分析与方法（AA）和应用与解释（AI）的考试都至关重要。

1. What is a Binomial Experiment? | 什么是二项试验？

A binomial experiment is a statistical experiment that satisfies four key properties: it consists of n repeated trials; each trial is independent of the others; each trial has exactly two possible outcomes, usually called success and failure; and the probability of success p is the same on every trial. The total number of successes X in these n trials is a binomial random variable, written as X ~ B(n, p).

二项试验是满足四个关键性质的统计试验：由n次重复试验组成；每次试验相互独立；每次试验只有两个可能结果，通常称为成功与失败；每次试验成功的概率p保持相同。这n次试验中成功总次数X是一个二项随机变量，记作X ~ B(n, p)。

For example, tossing a fair coin 10 times and counting the number of heads is a binomial experiment with n = 10 and p = 0.5. Another example is counting the number of defective items in a random sample of 20 products from a production line where 5% are defective.

例如，抛一枚均匀硬币10次并统计正面出现的次数就是一个二项试验，其中n = 10，p = 0.5。另一个例子是统计某生产线抽取20件产品中的次品数量，已知次品率为5%。

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2. The Binomial Probability Formula | 二项概率公式

The probability of obtaining exactly k successes in n trials is given by the binomial probability formula:

在n次试验中恰好得到k次成功的概率由二项概率公式给出：

P(X = k) = ₙCₖ pᵏ (1 − p)ⁿ⁻ᵏ

Here ₙCₖ is the binomial coefficient, equal to n! / (k!(n − k)!). This formula is often written as P(X = k) = C(n, k) pᵏ qⁿ⁻ᵏ, where q = 1 − p is the probability of failure on each trial.

其中ₙCₖ是二项系数，等于n! / (k!(n − k)!)。此公式也常写作P(X = k) = C(n, k) pᵏ qⁿ⁻ᵏ，其中q = 1 − p表示每次试验失败的概率。

For instance, if X ~ B(5, 0.4), then P(X = 3) = C(5, 3) × 0.4³ × 0.6² = 10 × 0.064 × 0.36 = 0.2304.

例如，若X ~ B(5, 0.4)，则P(X = 3) = C(5, 3) × 0.4³ × 0.6² = 10 × 0.064 × 0.36 = 0.2304。

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3. Conditions for Using the Binomial Distribution | 使用二项分布的条件

Before applying the binomial distribution, you must check that all of the following conditions hold:

在运用二项分布之前，必须确认以下条件全部满足：

• There is a fixed number of trials, n.

  试验次数n固定不变。

• Each trial is independent; the result of one trial does not affect any other trial.

  每次试验相互独立，一次试验的结果不会影响其他试验。

• Each trial has only two possible outcomes: success or failure.

  每次试验只有两个可能结果：成功或失败。

• The probability of success p is constant for every trial.

  每次试验成功的概率p保持不变。

If any of these conditions is not satisfied, the binomial model may be inappropriate and another distribution should be considered.

若以上任一条件不满足，二项模型可能不适用，需要考虑其他分布。

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4. Mean, Variance, and Standard Deviation | 均值、方差和标准差

For a binomial random variable X ~ B(n, p), the mean, variance, and standard deviation have very simple formulas:

对二项随机变量X ~ B(n, p)，其均值、方差和标准差有非常简洁的公式：

E(X) = μ = np

Var(X) = σ² = np(1 − p) = npq

SD(X) = σ = √(npq)

These formulas allow you to find the expected number of successes and the spread of the distribution without listing all probabilities.

这些公式能帮助你在不必逐一列出所有概率的情况下，直接求出期望成功次数以及分布的离散程度。

For example, if X ~ B(100, 0.2), then E(X) = 20, Var(X) = 100 × 0.2 × 0.8 = 16, and SD(X) = 4.

例如，若X ~ B(100, 0.2)，则E(X) = 20，Var(X) = 100 × 0.2 × 0.8 = 16，SD(X) = 4。

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5. Cumulative Binomial Probabilities | 累积二项概率

In many IB questions, you need to find the probability that X is at most k or at least k. The cumulative probability P(X ≤ k) is the sum of all individual probabilities from 0 up to k:

在许多IB题目中，需要求X至多为k或至少为k的概率。累积概率P(X ≤ k)是从0到k所有单个概率之和：

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