IB Mathematics: Binomial Distribution | IB数学：二项分布

📚 IB Mathematics: Binomial Distribution | IB数学：二项分布

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, where each trial has the same probability of success.

二项分布是IB数学中最重要的离散型概率分布之一。它用于建模在固定次数的独立试验中“成功”的次数，每次试验的成功概率保持不变。

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

In IB Mathematics, a random variable \(X\) is said to follow a binomial distribution if it counts the number of successes in \(n\) independent trials, each with success probability \(p\). This is written as \(X \sim B(n, p)\). The possible values of \(X\) are \(0, 1, 2, \dots, n\).

在IB数学中，如果一个随机变量 \(X\) 统计 \(n\) 次独立试验中成功的次数，且每次试验的成功概率为 \(p\)，则称 \(X\) 服从二项分布，记为 \(X \sim B(n, p)\)。\(X\) 的可能取值为 \(0, 1, 2, \dots, n\)。

For example, tossing a fair coin 10 times and counting the number of heads can be modelled by \(X \sim B(10, 0.5)\).

例如，抛一枚均匀硬币10次并统计正面朝上的次数，可以用 \(X \sim B(10, 0.5)\) 来建模。

2. Conditions for a Binomial Distribution | 二项分布的条件

Four essential conditions must be satisfied before a situation can be modelled by a binomial distribution:

在使用二项分布建模之前，必须满足四个基本条件：

Fixed number of trials \(n\) | 固定试验次数 \(n\)
Each trial has only two outcomes: success or failure | 每次试验只有两种结果：成功或失败
The probability of success \(p\) is the same for every trial | 每次试验的成功概率 \(p\) 保持不变
The trials are independent | 各次试验相互独立

If any of these conditions is violated, the binomial distribution is not appropriate.

如果以上任一条件不满足，则不宜使用二项分布。

3. Probability Mass Function | 概率质量函数

For \(X \sim B(n, p)\), the probability of observing exactly \(r\) successes is given by:

对于 \(X \sim B(n, p)\)，恰好观察到 \(r\) 次成功的概率为：

P(X = r) = nCr × pʳ × (1 − p)ⁿ⁻ʳ

Here, \(nCr\) is the binomial coefficient, which counts the number of ways to choose \(r\) successes from \(n\) trials. It can also be written as \(\frac{n!}{r!(n-r)!}\).

其中 \(nCr\) 是二项式系数，表示从 \(n\) 次试验中选出 \(r\) 次成功的方法数，也可写作 \(\frac{n!}{r!(n-r)!}\)。

For example, if \(X \sim B(5, 0.4)\), then \(P(X = 2) = 5C2 × 0.4² × 0.6³ = 10 × 0.16 × 0.216 = 0.3456\).

例如，若 \(X \sim B(5, 0.4)\)，则 \(P(X = 2) = 5C2 × 0.4² × 0.6³ = 10 × 0.16 × 0.216 = 0.3456\)。

4. Expected Value and Variance | 期望值与方差

The mean and variance of a binomial distribution have simple forms:

二项分布的均值和方差具有简洁的形式：

E(X) = np

Var(X) = np(1 − p)

The standard deviation is therefore \(\sqrt{np(1-p)}\). For example, if \(X \sim B(20, 0.3)\), then \(E(X) = 6\) and \(Var(X) = 20 × 0.3 × 0.7 = 4.2\).

因此标准差为 \(\sqrt{np(1-p)}\)。例如，若 \(X \sim B(20, 0.3)\)，则 \(E(X) = 6\)，\(Var(X) = 20 × 0.3 × 0.7 = 4.2\)。

5. Shape of the Distribution | 分布的形状

The shape of a binomial distribution depends on \(n\) and \(p\). When \(p = 0.5\), the distribution is symmetric. When \(p < 0.5\), the distribution is positively skewed (tail extends to the right). When \(p > 0.5\), it is negatively skewed (tail extends to the left).

二项分布的形状取决于 \(n\) 和 \(p\)。当 \(p = 0.5\) 时，分布对称；当 \(p < 0.5\) 时，分布呈正偏态（右尾较长）；当 \(p > 0.5\) 时，分布呈负偏态（左尾较长）。

As \(n\) increases, the distribution becomes more symmetric and approaches a normal distribution, especially when \(p\) is close to 0.5.

随着 \(n\) 增大，分布会变得更加对称，并逐渐趋近于正态分布，尤其当 \(p\) 接近0.5时更为明显。

6. Cumulative Probabilities | 累积概率

In IB examinations, questions often ask for \(P(X \le k)\), which is the cumulative probability from 0 to \(k\). This can be computed using a GDC or binomial cumulative distribution tables.

在IB考试中，经常需要计算 \(P(X \le k)\)，即从0到 \(k\) 的累积概率。这可以使用计算器或二项分布累积概率表来计算。

For intervals, use \(P(a \le X \le b) = P(X \le b) – P(X \le a-1)\). Also note that \(P(X < k) = P(X \le k-1)\).

对于区间概率，使用 \(P(a \le X \le b) = P(X \le b) – P(X \le a-1)\)。同时注意 \(P(X < k) = P(X \le k-1)\)。

7. Using a Graphic Display Calculator (GDC) | 使用图形计算器

On a TI-84 calculator, the command binompdf calculates \(P(X = r)\) and binomcdf calculates \(P(X \le k)\). For example, binompdf(10,0.3,3) gives the probability of exactly 3 successes.

在TI-84计算器上，binompdf 命令计算 \(P(X = r)\)，binomcdf 命令计算 \(P(X \le k)\)。例如 binompdf(10,0.3,3) 会给出恰好3次成功的概率。

On a Casio calculator, open the STAT menu, select DIST, then BINM, and choose Bpd for individual probabilities or Bcd for cumulative probabilities.

在Casio计算器上，进入STAT菜单，选择DIST，然后BINM，再选择Bpd计算单点概率，或选择Bcd计算累积概率。

In IB exams, showing calculator commands and writing the result to 3 significant figures is usually acceptable.

在IB考试中，写出计算器命令并将结果保留到3位有效数字通常是可接受的。

8. Real-World Applications | 实际应用

The binomial distribution appears in many real-world contexts. In quality control, it models the number of defective items in a batch. In medicine, it models the number of patients who recover after a treatment. In sports, it can model the number of successful free throws in basketball.

二项分布在许多现实情境中都有应用。在质量控制中，它可以模拟一批产品中的次品数量；在医学中，它可以模拟治疗后康复的患者人数；在体育中，它可以模拟篮球罚球命中的次数。

For example, if a basketball player has a free-throw success rate of 0.8 and takes 12 free throws, the number of successful shots can be modelled by \(B(12, 0.8)\).

例如，如果一位篮球运动员的罚球命中率为0.8，他罚球12次，那么命中次数可以用 \(B(12, 0.8)\) 来建模。

9. Normal Approximation to the Binomial | 二项分布的正态近似

When \(n\) is large, the binomial distribution can be approximated by a normal distribution. The rule of thumb is that both \(np\) and \(n(1-p)\) should be greater than 5. The approximating distribution is:

当 \(n\) 较大时，二项分布可以用正态分布来近似。经验法则是 \(np\) 和 \(n(1-p)\) 都应大于5。近似分布为：

X ≈ Normal(np, np(1 − p))

A continuity correction is needed because the binomial distribution is discrete and the normal distribution is continuous. If \(X \sim B(n,p)\) and we approximate by \(Y \sim Normal(np, np(1-p))\), then \(P(X \le k) \approx P(Y \le k + 0.5)\).

由于二项分布是离散分布而正态分布是连续分布，所以需要进行连续性修正。若 \(X \sim B(n,p)\) 且用 \(Y \sim Normal(np, np(1-p))\) 近似，则 \(P(X \le k) \approx P(Y \le k + 0.5)\)。

10. Binomial Distribution in Hypothesis Testing | 假设检验中的二项分布

In IB Mathematics, the binomial distribution is often used in hypothesis tests for a proportion. The null hypothesis is written as \(H_0: p = p_0\), where \(p_0\) is a claimed probability of success.

在IB数学中，二项分布常用于关于比例的假设检验。原假设写作 \(H_0: p = p_0\)，其中 \(p_0\) 是所声称的成功概率。

The alternative hypothesis can be one-tailed (\(H_1: p < p_0\) or \(H_1: p > p_0\)) or two-tailed (\(H_1: p \neq p_0\)). The test statistic is the observed number of successes in \(n\) trials.

备择假设可以是单尾的（\(H_1: p < p_0\) 或 \(H_1: p > p_0\)），也可以是双尾的（\(H_1: p \neq p_0\)）。检验统计量是在 \(n\) 次试验中观察到的成功次数。

For example, to test whether a coin is biased towards heads, we might toss it 20 times and count heads. Under \(H_0: p = 0.5\), \(X \sim B(20, 0.5)\). If the observed \(X\) is very large, this provides evidence against \(H_0\).

例如，要检验一枚硬币是否偏向正面，可以抛20次并统计正面次数。在 \(H_0: p = 0.5\) 下，\(X \sim B(20, 0.5)\)。如果观察到的 \(X\) 非常大，则为拒绝 \(H_0\) 提供了证据。

11. Common Mistakes | 常见错误

Students often confuse \(p\) and \(1-p\). Always identify which outcome is defined as “success” before applying the formula.

学生常将 \(p\) 与 \(1-p\) 混淆。应用公式前一定要先明确哪种结果被定义为“成功”。

Another common mistake is using the wrong inequality sign in cumulative probabilities. Remember that \(P(X < k)\) does not equal \(P(X \le k)\); it equals \(P(X \le k-1)\).

另一个常见错误是在累积概率中使用错误的不等号。记住 \(P(X < k)\) 不等于 \(P(X \le k)\)，而是等于 \(P(X \le k-1)\)。

Also, do not use the binomial distribution if the trials are not independent or if the probability of success changes from trial to trial.

此外，如果试验不独立或成功概率每次变化，则不要使用二项分布。

12. Summary and Revision Tips | 总结与复习要点

The binomial distribution is a powerful tool for modelling discrete random variables. Key formulas to remember are:

二项分布是离散型随机变量建模的有力工具。需要记住的关键公式有：

P(X = r) = nCr × pʳ × (1 − p)ⁿ⁻ʳ

E(X) = np

Var(X) = np(1 − p)

When revising, practise identifying binomial situations, using GDC commands, and performing hypothesis tests with the binomial distribution. Always check the four conditions before starting a problem.

复习时，应练习识别二项分布情境、使用计算器命令以及利用二项分布进行假设检验。解题前务必检查四个条件是否满足。

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IB Mathematics: Binomial Distribution | IB数学：二项分布