A-Level Mathematics: Conditions and Probability Calculations for the Binomial Distribution | A-Level数学：二项分布的条件与概率计算

📚 A-Level Mathematics: Conditions and Probability Calculations for the Binomial Distribution | A-Level数学：二项分布的条件与概率计算

The binomial distribution is one of the most important discrete probability distributions in A-Level Mathematics. It models the number of successes in a fixed number of independent trials, and it appears in many examination questions involving probability and hypothesis testing.

二项分布是A-Level数学中最重要的离散概率分布之一。它用于建模固定次数独立试验中的成功次数，并且出现在许多涉及概率和假设检验的考试题目中。

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

A binomial distribution describes the number of successes in a fixed number of independent trials, each with the same probability of success. We write X ~ B(n, p), where n is the number of trials and p is the probability of success on each trial.

二项分布描述的是在固定次数的独立试验中成功出现的次数，每次试验的成功概率相同。我们记作 X ~ B(n, p)，其中 n 是试验次数，p 是每次试验成功的概率。

For example, if a fair coin is tossed 10 times, the number of heads follows B(10, 0.5).

例如，将一枚均匀硬币抛掷10次，出现正面的次数服从 B(10, 0.5)。

The random variable X is discrete and can take integer values from 0 to n.

随机变量 X 是离散型的，可以取从 0 到 n 的整数。

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2. The Four Conditions | 二项分布的四个条件

Before using a binomial model, you must check that all four conditions are satisfied.

在使用二项模型之前，你必须检查四个条件是否全部满足。

• Fixed number of trials: the experiment is repeated exactly n times.

  固定试验次数：试验恰好重复 n 次。

• Each trial is independent: the outcome 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。

• Two possible outcomes: each trial can be classified as either a success or a failure.

  只有两种可能结果：每次试验只能被归类为成功或失败。

• Constant probability of success: the probability p of success is the same in every trial.

  成功概率恒定不变：每次试验成功的概率 p 都相同。

If any of these conditions is not met, the binomial distribution may not be appropriate.

如果这些条件中有任何一条不满足，二项分布可能就不适用。

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3. The Probability Mass Function (PMF) | 概率质量函数

If X ~ B(n, p), the probability of getting exactly r successes is given by the probability mass function:

若 X ~ B(n, p)，恰好得到 r 次成功的概率由概率质量函数给出：

P(X = r) = C(n, r) pʳ (1 − p)ⁿ⁻ʳ

Here C(n, r) counts the number of ways to choose r successes from n trials. The possible values of r are 0, 1, 2, …, n.

其中 C(n, r) 表示从 n 次试验中选出 r 次成功的方法数。r 的可能取值为 0, 1, 2, …, n。

For example, if X ~ B(10, 0.3), then P(X = 4) = C(10, 4) × 0.3⁴ × 0.7⁶.

例如，若 X ~ B(10, 0.3)，则 P(X = 4) = C(10, 4) × 0.3⁴ × 0.7⁶。

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4. Understanding Combinations ⁿCᵣ | 理解组合数 ⁿCᵣ

The binomial coefficient C(n, r) is also written as ⁿCᵣ or “n choose r”. It represents the number of ways to select r items from n items without regard to order.

二项系数 C(n, r) 也写作 ⁿCᵣ 或“n 选 r”。它表示从 n 个元素中不考虑顺序地选取 r 个元素的方法数。

C(n, r) = n! / (r! (n − r)!)

For example, C(10, 4) = 10! / (4! × 6!) = (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1) = 210.

例如，C(10, 4) = 10! / (4! × 6!) = (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1) = 210。

Notice that C(n, r) = C(n, n − r), which is useful when simplifying calculations.

注意 C(n, r) = C(n, n − r)，这在简化计算时很有用。

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5. Calculating Individual Probabilities | 计算单个概率

To find P(X = r), substitute the values of n, p and r into the PMF or use your calculator’s binomial probability function.

要求 P(X = r)，将 n、p 和 r 的值代入概率质量函数，或使用计算器的二项分布概率功能。

Worked example: Let X ~ B(10, 0.3). Find P(X = 4).

示例：设 X ~ B(10, 0.3)，求 P(X = 4)。

P(X = 4) = 210 × 0.3⁴ × 0.7⁶ ≈ 210 × 0.0081 × 0.117649 ≈ 0.2001

So P(X = 4) ≈ 0.2001, or about 20.0%.

因此 P(X = 4) ≈ 0.2001，即约20.0%。

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6. Cumulative Probabilities: P(X ≤ r) and P(X ≥ r) | 累积概率：P(X ≤ r) 与 P(X ≥ r)

Examination questions often ask for P(X ≤ r) or P(X ≥ r). These are found by adding individual binomial probabilities.

考试题目经常要求 P(X ≤ r) 或 P(X ≥ r)。这些可以通过累加单个二项概率得到。

P(X ≤ r) = P(X = 0) + P(X = 1) + … + P(X = r)

Using the complement rule, P(X ≥ r) = 1 − P(X ≤ r − 1). In particular, P(X ≥ 1) = 1 − P(X = 0).

利用补事件规则，P(X ≥ r) = 1 − P(X ≤ r − 1)。特别地，P(X ≥ 1) = 1 − P(X = 0)。

Example: For X ~ B(10, 0.3), P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2) ≈ 0.0282 + 0.1211 + 0.2335 = 0.3828.

例如：对于 X ~ B(10, 0.3)，P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2) ≈ 0.0282 + 0.1211 + 0.2335 = 0.3828。

Also, P(X ≥ 1) = 1 − 0.7¹⁰ ≈ 0.9718.

同时，P(X ≥ 1) = 1 − 0.7¹⁰ ≈ 0.9718。

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7. Mean and Variance | 期望与方差

For a binomial distribution X ~ B(n, p), the mean and variance have simple formulas.

对于二项分布 X ~ B(n, p)，其期望与方差有简单的公式。

E(X) = np, Var(X) = np(1 − p)

The standard deviation is the square root of the variance: SD(X) = √(np(1 − p)).

标准差是方差的正平方根：SD(X) = √(np(1 − p))。

For B(10, 0.3), E(X) = 10 × 0.3 = 3 and Var(X) = 10 × 0.3 × 0.7 = 2.1.

对于 B(10, 0.3)，E(X) = 10 × 0.3 = 3，Var(X) = 10 × 0.3 × 0.7 = 2.1。

These results are useful for quick checks and for solving problems about expected values.

这些结果可用于快速检验，也可用于解决与期望值相关的问题。

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8. Using Calculator Functions | 使用计算器功能

Most modern calculators have a binomial distribution function. For X ~ B(n, p), you can usually enter n, p and the required value of r to obtain P(X = r) or P(X ≤ r).

大多数现代计算器都有二项分布功能。对于 X ~ B(n, p)，你通常可以输入 n、p 和所需的 r 值，得到 P(X = r) 或 P(X ≤ r)。

To find P(X ≥ r), calculate 1 − P(X ≤ r − 1) because most calculators give cumulative probabilities from 0 directly.

要求 P(X ≥ r)，计算 1 − P(X ≤ r − 1)，因为大多数计算器直接给出从 0 开始的累积概率。

Be careful with the boundary: P(X < r) means P(X ≤ r − 1), while P(X ≤ r) includes r itself.

要注意边界：P(X < r) 表示 P(X ≤ r − 1)，而 P(X ≤ r) 包含 r 本身。

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9. Conditional Probability with a Binomial Distribution | 二项分布中的条件概率

Conditional probability measures the probability of one event occurring given that another event has already occurred. The general formula is:

条件概率衡量的是在另一个事件已经发生的条件下，某个事件发生的概率。其通用公式为：

P(A | B) = P(A ∩ B) / P(B)

In binomial questions, a common condition is “given that there is at least one success”, i.e. X ≥ 1.

在二项分布题目中，常见的条件是“已知至少有一次成功”，即 X ≥ 1。

If k ≥ 1, then the event X = k automatically implies X ≥ 1, so P(X = k ∩ X ≥ 1) = P(X = k). Therefore:

若 k ≥ 1，则事件 X = k 本身就意味着 X ≥ 1，所以 P(X = k ∩ X ≥ 1) = P(X = k)。因此：

P(X = k | X ≥ 1) = P(X = k) / P(X ≥ 1) = P(X = k) / (1 − P(X = 0))

For X ~ B(10, 0.3), P(X = 3 | X ≥ 1) = P(X = 3) / (1 − P(X = 0)) ≈ 0

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