📚 A-Level Maths: Cumulative Probability in Binomial Distribution | A-Level 数学：二项分布中的累积概率计算
In A-Level Mathematics, the binomial distribution is one of the most frequently tested discrete probability distributions. Many exam questions go beyond a single probability and ask for cumulative probabilities, such as P(X ≤ x) or P(X > x). Mastering this skill is essential for scoring well.
在 A-Level 数学中，二项分布是最常考的离散型概率分布之一。许多考试题目不止要求计算单一概率，还会要求累积概率，例如 P(X ≤ x) 或 P(X > x)。掌握这一技能对取得高分至关重要。
1. Review of Binomial Distribution | 二项分布回顾
A binomial random variable X counts the number of successes in n independent trials, where each trial has the same probability of success p. We write X ~ B(n, p). There are only two outcomes in each trial: success or failure, and the probability of failure is q = 1 − p.
二项随机变量 X 用于统计 n 次独立试验中的成功次数，每次试验的成功概率均为 p。我们记作 X ~ B(n, p)。每次试验只有两种结果：成功或失败，失败概率为 q = 1 − p。
The probability mass function gives the probability of exactly r successes:
概率质量函数给出恰好 r 次成功的概率：
P(X = r) = C(n, r) pʳ(1−p)ⁿ⁻ʳ
Here C(n, r) is the binomial coefficient, sometimes written as ⁿCᵣ or ‘n choose r’. For example, X ~ B(10, 0.3) means 10 independent trials with success probability 0.3.
其中 C(n, r) 是二项系数，有时记作 ⁿCᵣ 或“n 选 r”。例如，X ~ B(10, 0.3) 表示进行 10 次独立试验，每次成功概率为 0.3。
2. What Is Cumulative Probability? | 什么是累积概率？
A cumulative probability is the probability that X takes a value less than or equal to a given number x. It is written as P(X ≤ x) and is often called the cumulative distribution function, or CDF.
累积概率是指 X 取小于或等于给定数值 x 的概率，记作 P(X ≤ x)，也常称为累积分布函数（CDF）。
The CDF is found by adding all individual probabilities from the smallest possible value up to x:
CDF 的计算方法是将从最小可能值到 x 的所有单个概率相加：
F(x) = P(X ≤ x) = Σ C(n, r) pʳ(1−p)ⁿ⁻ʳ, for r = 0 to x
For a binomial distribution, the smallest possible value is always 0, so the sum always starts at r = 0.
对于二项分布，最小可能值始终为 0，因此求和总是从 r = 0 开始。
3. Key Cumulative Probability Rules | 累积概率的关键转换规则
Exam questions often ask for probabilities such as P(X < x), P(X > x) or P(X ≥ x). You must be able to convert these into the standard form P(X ≤ k).
考试题常常要求 P(X < x)、P(X > x) 或 P(X ≥ x) 这样的概率。你必须能够将这些形式转换为标准形式 P(X ≤ k)。
The following rules are essential:
以下转换规则至关重要：
- P(X < r) = P(X ≤ r − 1)
- P(X ≤ r) is given directly by tables or calculator
- P(X > r) = 1 − P(X ≤ r)
- P(X ≥ r) = 1 − P(X ≤ r − 1)
- P(a ≤ X ≤ b) = P(X ≤ b) − P(X ≤ a − 1)
Notice that P(X > r) and P(X ≥ r) are not the same. The first excludes r, while the second includes r. Always check whether the boundary value is included.
注意 P(X > r) 与 P(X ≥ r) 并不相同：前者不包括 r，后者包括 r。做题时务必检查边界值是否被包含。
4. Using Statistical Tables | 使用统计表
In the exam, you may be given a table of cumulative binomial probabilities. These tables usually show P(X ≤ x) for selected values of n and p. The row gives x, and the column gives p.
考试中可能会提供二项分布累积概率表。这类表通常给出选定 n 和 p 下的 P(X ≤ x)，行对应 x，列对应 p。
For example, for n = 10 and p = 0.3, the table may show:
例如，当 n = 10、p = 0.3 时，表中可能显示：
| x | 3 | 4 | 5 |
| P(X ≤ x) | 0.6496 | 0.8497 | 0.9527 |
So P(X ≤ 3) = 0.6496 directly. If the table is incomplete, you may need to subtract two entries to find an intermediate probability.
因此可以直接读出 P(X ≤ 3) = 0.6496。如果表格不完整，可能需要用两个条目的差值来求中间区间的概率。
5. Using a Calculator | 使用计算器
Modern scientific calculators can compute binomial cumulative probabilities directly. On many models, look for a Distribution menu and choose ‘Binomial C.D.’ or ‘binomCdf’.
现代科学计算器可以直接计算二项分布累积概率。许多型号可在“分布”菜单中选择“Binomial C.D.”或“binomCdf”。
For an interval, you can enter lower and upper boundaries. For example, X ~ B(10, 0.3), to find P(X ≤ 3), set lower = 0, upper = 3. The calculator will return 0.6496107…
对于区间，可以输入下限和上限。例如 X ~ B(10, 0.3)，要求 P(X ≤ 3)，可设下限为 0、上限为 3，计算器会返回 0.6496107…。
Using the calculator is faster and reduces arithmetic errors, but you must still know which boundaries to use for expressions like P(X < x) or P(X ≥ x).
使用计算器更快且能减少算术错误，但你仍然需要知道对于 P(X < x) 或 P(X ≥ x) 等表达式应使用什么边界。
6. Worked Example 1: Finding P(X ≤ r) | 实例 1：求 P(X ≤ r)
Let X ~ B(10, 0.3). Find P(X ≤ 4).
设 X ~ B(10, 0.3)，求 P(X ≤ 4)。
Using tables or a calculator, the cumulative probability is P(X ≤ 4) = 0.8497. This means there is about an 85% chance of at most 4 successes in 10 trials.
使用统计表或计算器，累积概率为 P(X ≤ 4) = 0.8497。这意味着在 10 次试验中最多出现 4 次成功的概率约为 85%。
If you need P(X < 4), use the rule: P(X < 4) = P(X ≤ 3) = 0.6496.
如果需要 P(X < 4)，利用规则：P(X < 4) = P(X ≤ 3) = 0.6496。
7. Worked Example 2: Finding P(X ≥ r) | 实例 2：求 P(X ≥ r)
Still using X ~ B(10, 0.3), find P(X ≥ 6).
仍然设 X ~ B(10, 0.3)，求 P(X ≥ 6)。
Since P(X ≥ 6) = 1 − P(X ≤ 5), we first find P(X ≤ 5) from the table. For n = 10, p = 0.3, P(X ≤ 5) = 0.9527.
因为 P(X ≥ 6) = 1 − P(X ≤ 5)，我们先从表中查找 P(X ≤ 5)。当 n = 10、p = 0.3 时，P(X ≤ 5) = 0.9527。
P(X ≥ 6) = 1 − 0.9527 = 0.0473
So the probability of 6 or more successes is only about 4.7%. Notice how we did not need to add P(X = 6), P(X = 7), etc. separately.
因此，出现 6 次或更多成功的概率仅为约 4.7%。注意我们不需要分别相加 P(X = 6)、P(X = 7) 等。
8. Worked Example 3: Finding P(a ≤ X ≤ b) | 实例 3：求区间概率 P(a ≤ X ≤ b)
Find P(2 ≤ X ≤ 5) for X ~ B(10, 0.3).
对于 X ~ B(10, 0.3)，求 P(2 ≤ X ≤ 5)。
Use the interval rule: P(2 ≤ X ≤ 5) = P(X ≤ 5) − P(X ≤ 1).
使用区间规则：P(2 ≤ X ≤ 5) = P(X ≤ 5) − P(X ≤ 1)。
From the table, P(X ≤ 5) = 0.9527 and P(X ≤ 1) = 0.1493. Therefore:
由表可知，P(X ≤ 5) = 0.9527，P(X ≤ 1) = 0.1493。因此：
P(2 ≤ X ≤ 5) = 0.9527 − 0.1493 = 0.8034
A common mistake is to subtract P(X ≤ 2) instead of P(X ≤ 1). Since 2 is included in the interval, you must subtract all values below 2.
常见错误是用 P(X ≤ 2) 相减，而应该用 P(X ≤ 1)。因为 2 包含在区间内，所以必须减去所有小于 2 的值。
9. Inverse Problem: Finding Critical Values | 逆问题：求临界值
Sometimes you are given the cumulative probability and asked to find the value of x. For example, find the smallest x such that P(X ≤ x) ≥ 0.95, where X ~ B(10, 0.3).
有时题目会给出累积概率，要求求出 x 的值。例如，设 X ~ B(10, 0.3)，求满足 P(X ≤ x) ≥ 0.95 的最小 x。
Check the cumulative table in increasing x order:
按 x 递增的顺序检查累积概率表：
| x | 3 | 4 | 5 |
| P(X ≤ x) | 0.6496 | 0.8497 | 0.9527 |
Since P(X ≤ 4) = 0.8497 < 0.95 and P(X ≤ 5) = 0.9527 ≥ 0.95, the required value is x = 5. This value is sometimes called the critical value or quantile.
由于 P(X ≤ 4) = 0.8497 < 0.95，而 P(X ≤ 5) = 0.9527 ≥ 0.95，所以要求的值为 x = 5。这个值有时称为临界值或分位数。
10. Common Mistakes and Pitfalls | 常见错误与陷阱
The most common exam errors in binomial cumulative probability are listed below.
在二项分布累积概率计算中，最常见的考试错误如下所列。
- Confusing P(X < r) with P(X ≤ r). Remember: P(X < r) = P(X ≤ r − 1).
- 计算中混淆 P(X < r) 与 P(X ≤ r)。记住：P(X < r) = P(X ≤ r − 1)。
- Using the wrong p or n. Always check whether p is the probability of success, not failure.
- 使用了错误的 p 或 n。务必检查 p 是成功概率，而不是失败概率。
- Forgetting that the sum of all probabilities is 1, so P(X ≥ r) = 1 − P(X ≤ r − 1).
- 忘记所有概率之和为 1，因此 P(X ≥ r) = 1 − P(X ≤ r − 1)。
- Using a normal approximation without checking the conditions np > 5 and n(1−p) > 5. For the binomial exact cumulative probability, tables or the calculator should be used when possible.
- 未检查 np > 5 且 n(1−p) > 5 就使用正态近似。对于二项精确累积概率，应尽可能使用统计表或计算器。
- When p > 0.5, tables for small p may not apply directly. You can transform to Y = n − X, where Y ~ B(n, 1 − p).
- 当 p > 0.5 时，小 p 的表格可能不适用。你可以转换 Y = n − X，此时 Y ~ B(n, 1 − p)。
11. Exam Tips and Summary | 考试技巧与小结
Always write down the distribution clearly: X ~ B(n, p). Then decide whether the question wants P(X ≤ x), P(X > x), or P(a ≤ X ≤ b).
做题时先清楚写出分布：X ~ B(n, p)。然后判断题目要求的是 P(X ≤ x)、P(X > x) 还是 P(a ≤ X ≤ b)。
If the question gives a table, use it directly. If you use a calculator, write down the values you obtain at each step so that you earn method marks even if the final answer is wrong.
如果题目提供统计表，可直接使用。如果使用计算器，请在每一步写下所得的数值，这样即使最终结果有误，也能获得方法分。
Check boundary conditions: ‘more than 5’ means X > 5, so use 1 − P(X ≤ 5); ‘at least 5’ means X ≥ 5, so use 1 − P(X ≤ 4).
注意边界条件：“more than 5”表示 X > 5，应用 1 − P(X ≤ 5)；“at least 5”表示 X ≥ 5，应用 1 − P(X ≤ 4)。
Finally, practise converting between the four forms: P(X ≤ r), P(X < r), P(X > r)
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