🎲 古典概型：数一数就知道概率🎲 Classical Probability: Just Count

上一片我们认识了随机事件：结果说不准，事先不知道会怎样。但有一类情形能算得清清楚楚——只要每个结果机会一样，数一数就知道概率。 Last leaf we met random events — outcomes you can't predict in advance. But one big family of them can be pinned down exactly: when every outcome is equally likely, you just count.

第 1 步：袋子里有 5 个球Step 1: Five balls in a jar

先看最简单的实验：从袋子里摸一个球。Start with the simplest experiment: draw one ball from the jar.

3 红 2 蓝的袋子和 5 种结果的清单 — 小算盘机器人准备了 1 个透明罐：里面 3 个红球、2 个蓝球，大小重量一模一样。每个球被摸到的机会完全一样，这叫**等可能**。把所有可能的结果排出来，正好 5 种——红球占了其中 3 种。 The little abacus robot fills a clear jar with 3 red balls and 2 blue ones — same size, same weight. Every ball has exactly the same chance of being drawn: that's **equally likely**. Lay all outcomes in a row and you count 5 — and 3 of them are red.

数出了结果，概率怎么算？Now that we can count, how do we turn it into a probability?

第 2 步：概率 = 有利结果 ÷ 全部结果Step 2: Probability = wanted outcomes ÷ all outcomes

骰子六面：偶数 2、4、6 共 3 面 — 拿骰子练手：6 个面机会一样，偶数有 2、4、6 共 3 面，所以 P(偶数) = 3 ÷ 6 = 1/2。套回袋子：P(红) = 3 ÷ 5。分子是“你要的结果数”，分母是“全部结果数”——这就是古典概型的全部秘密。 Practice with a die: all 6 faces are equally likely, and the even faces are 2, 4 and 6 — three of them, so P(even) = 3 ÷ 6 = 1/2. Back to the jar: P(red) = 3 ÷ 5. Numerator: the outcomes you want. Denominator: all outcomes. That is the whole secret of classical probability.

那这个“数一数”的办法，什么时候会失灵？So when does counting stop working?

第 3 步：前提是“每份一样大”Step 3: The fine print — “equally likely”

5 等分的转盘：3 红 2 蓝 — 抽奖转盘平均分成 5 格（3 红 2 蓝），指针停在每格的机会一样，照样数：P(红) = 3 ÷ 5。可要是格子有大有小，或者球有轻有重，**等可能**就破了——那时数出来的比例会骗人。用这招前，先确认：每一份真的一样可能。 A prize wheel split into 5 equal slices (3 red, 2 blue): the pointer is equally likely to stop on any slice, so count again: P(red) = 3 ÷ 5. But if the slices had different sizes, or some balls were heavier, **equally likely** breaks — and the counting would lie. Before using this trick, check: is every outcome really equally likely?

🎮 摸球实验：先猜，再摸 100 次🎮 The drawing lab: guess first, then draw 100 times

道理懂了，来验证：先选出正确答案，再让机器人真的摸 100 次，看频率怎么靠近概率。Time to test it: pick the right probability, then let the robot really draw 100 times and watch the frequency approach it.

一句话记住它：古典概型 = 等可能的前提下，概率 = 有利结果数 ÷ 全部结果数。 Remember it in one line: classical probability — when outcomes are equally likely, P = wanted outcomes ÷ all outcomes.

等可能是前提：球一样大、格子一样宽，机会才一样“Equally likely” comes first: same-size balls, same-width slices, same chance 概率是 0~1 之间的比例，不是“肯定发生”A probability is a ratio between 0 and 1 — not a guarantee 实验次数越多，频率越贴近理论概率The more trials, the closer the experimental frequency gets to the theoretical probability

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