Binomial 二项分布计算器（单点、累积概率与分布表）Binomial Distribution Calculator (Exact, Cumulative & Table)

使用说明与常见问题How-to & FAQ

「二项分布计算器（单点、累积概率与分布表）」是什么？What is the Binomial Distribution Calculator (Exact, Cumulative & Table) tool?

由试验次数 n、成功概率 p 和成功次数 k 算 P(X = k)、P(X ≤ k)、P(X < k)、P(X ≥ k)、P(X > k)，以及均值 np、方差和标准差；n 不大时列出整张分布表。n 最多 10,000。本地处理。From the number of trials n, the success probability p and the number of successes k, get P(X = k), P(X ≤ k), P(X < k), P(X ≥ k) and P(X > k), plus the mean np, variance and standard deviation, and the full distribution table for small n. n up to 10,000. Processed locally.

如何使用「二项分布计算器（单点、累积概率与分布表）」？How do I use the Binomial Distribution Calculator (Exact, Cumulative & Table) tool?

在「二项分布计算器（单点、累积概率与分布表）」中，填写「试验次数 n」、「每次成功概率 p（0–1）」、「成功次数 k」等参数，结果会随输入实时计算并显示在下方。In the Binomial Distribution Calculator (Exact, Cumulative & Table): Fill in "Number of trials n", "Success probability p (0–1)", "Number of successes k"; results are recalculated instantly as you type.

什么时候用二项分布？When does the binomial distribution apply?

满足四个条件：试验次数 n 固定；每次只有成功、失败两种结果；每次成功概率 p 相同；各次试验相互独立。例如抛 10 次硬币数正面。When four things hold: a fixed number of trials n; each trial is a success or failure; the success probability p is the same every time; and the trials are independent. Counting heads in 10 coin tosses is an example.

“至少”和“最多”怎么选？Which one is at least and which is at most?

“最多 k 次”是 P(X ≤ k)，“少于 k 次”是 P(X < k)，“至少 k 次”是 P(X ≥ k)，“多于 k 次”是 P(X > k)。页面五个都列出来。At most k is P(X ≤ k), fewer than k is P(X < k), at least k is P(X ≥ k) and more than k is P(X > k). All five are listed.

n 很大时准吗？Is it accurate for large n?

准。页面用对数计算阶乘，不会溢出，n 可到 10,000，不用正态近似。Yes. Factorials are computed with logarithms, so nothing overflows up to n = 10,000 and no normal approximation is used.

能举个计算实例吗？Can you show a worked example?

例如在「二项分布计算器（单点、累积概率与分布表）」中填 试验次数 n = 5，每次成功概率 p（0–1） = 0.5，成功次数 k = 2，结果区显示：P(X = 2)：0.3125；P(X ≤ 2)：0.5；P(X < 2)：0.1875；P(X ≥ 2)：0.8125；P(X > 2)：0.5；均值 μ = np：2.5；方差 σ² = np(1 − p)：1.25；标准差 σ：1.118034；ℹ P(X = k) = C(n, k)·p^k·(1 − p)^(n − k)For example, entering Number of trials n = 5, Success probability p (0–1) = 0.5, Number of successes k = 2 in the Binomial Distribution Calculator (Exact, Cumulative & Table) gives: P(X = 2): 0.3125; P(X ≤ 2): 0.5; P(X < 2): 0.1875; P(X ≥ 2): 0.8125; P(X > 2): 0.5; Mean μ = np: 2.5; Variance σ² = np(1 − p): 1.25; Standard deviation σ: 1.118034; ℹ P(X = k) = C(n, k)·p^k·(1 − p)^(n − k)

哪些输入会被检查？Which inputs are validated?

「二项分布计算器（单点、累积概率与分布表）」会检查这些输入——「试验次数 n」：必须填写；「每次成功概率 p（0–1）」：必须填写；「成功次数 k」：必须填写，不能为负数，须为整数；绝对值超过 10¹⁵ 的数也会被拒绝。不符合时结果区显示具体提示，不会给出数字。The Binomial Distribution Calculator (Exact, Cumulative & Table) checks its inputs — "Number of trials n": required; "Success probability p (0–1)": required; "Number of successes k": required, cannot be negative, must be a whole number; numbers above 10¹⁵ in absolute value are rejected too. When a value fails, the result area names the problem instead of showing a number.

有哪些选项可以调整？What options can I adjust?

「二项分布计算器（单点、累积概率与分布表）」的设置：「试验次数 n」默认 10；「每次成功概率 p（0–1）」默认 0.5；「成功次数 k」默认 3。每次生成结果都按当前设置处理。Settings in the Binomial Distribution Calculator (Exact, Cumulative & Table) — "Number of trials n" default 10; "Success probability p (0–1)" default 0.5; "Number of successes k" default 3. The result always uses the current settings.

「二项分布计算器（单点、累积概率与分布表）」免费吗？需要注册吗？Is the Binomial Distribution Calculator (Exact, Cumulative & Table) free? Do I need an account?

「二项分布计算器（单点、累积概率与分布表）」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Binomial Distribution Calculator (Exact, Cumulative & Table) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.