🔍 条件概率：知道了新消息之后🔍 Conditional Probability: After the News
上一片我们数出 P(红) = 3/5。现在有人递来一条新消息：“你摸到的球不是蓝球。” 概率会变吗？会——而且变得有理有据。Last leaf we counted P(red) = 3/5. Now someone whispers a piece of news: “Your ball is not blue.” Does the probability change? It does — and it changes in a way you can count.
第 1 步：新消息让可能性变少Step 1: News shrinks the possibilities
先看“可能性”本身发生了什么事。First, watch what happens to the possibilities themselves.
本来有 5 种可能，一视同仁。消息“不是蓝球”一进来，2 个蓝球当场出局——分母从 5 缩成 3。这就是条件概率：拿到新消息后，把世界缩小，再重新数一遍。There were 5 equally likely outcomes. The moment the news “not blue” arrives, the 2 blue balls are out — the denominator shrinks from 5 to 3. That's conditional probability: take the news, shrink the world, count again.
缩小之后，“我要的那一个”怎么数？Once the range is smaller, how do we count the one we want?
第 2 步：在缩小后的范围里数Step 2: Count inside the shrunken range
4 张牌摆桌上：红桃K、红桃A、黑桃K、黑桃A。消息“抽到的是 K”→ 只剩 2 张 K。在这 2 张里，红桃 K 占 1 张，所以 P(红桃K | 是K) = 1 ÷ 2。竖线读作“在……条件下”：先有 K，再谈红桃。Four cards on the table: heart K, heart A, spade K, spade A. News: “the card is a K” → only 2 kings remain. Among those 2, exactly 1 is the heart K, so P(heart K | it's a K) = 1 ÷ 2. The vertical bar reads “given”: given it's a king, how likely is a heart?
换一个更生活的场景：出门带伞。Now a more everyday scene: taking an umbrella.
第 3 步：下雨天再看带伞Step 3: Rainy days, and umbrellas
机器人记了 10 天天气：其中 4 天下雨。问“带伞的概率”：不加条件，在 10 天里数；加上“已知这天下了雨”，就只在 4 个雨天里数——3 天带了伞，P(带伞 | 下雨) = 3 ÷ 4。条件换了，分母就换了。The robot keeps 10 days of weather records: 4 rainy days. Ask about umbrellas: with no news, you count across all 10 days; with “given it rained”, you count only inside the 4 rainy days — 3 of them had an umbrella, so P(umbrella | rain) = 3 ÷ 4. Change the condition, change the denominator.
🎮 缩小样本空间：闯 4 关🎮 Shrink the sample space: 4 levels
每关先读一条新消息，把不符合的可能划掉，再在剩下的里面选答案。Each level hands you a piece of news — cross out the outcomes that don't match, then choose your answer from what's left.
一句话记住它：P(A|B) = 已知 B 发生时 A 的概率：先把范围缩到 B，再在 B 里数 A。Remember it in one line: P(A|B) is the probability of A once you know B happened: shrink the range to B, then count A inside B.
竖线“|”读作“在……条件下”，后面跟着已知的新消息The bar “|” reads “given” — what follows it is the news you already have新消息把分母变小：不符合条件的结果全部出局News shrinks the denominator: outcomes that fail the condition drop out同一件事，条件不同概率可能不同：P(红桃K)=1/4，但 P(红桃K｜是K)=1/2Same event, different condition, different probability: P(heart K)=1/4, but P(heart K | it's a K)=1/2
内容参考 OpenStax 等公开教材，多来源核对 · AI 生成、人工审核 · 发现错误欢迎指正，帮这片叶子长得更好。Based on OpenStax and other open textbooks, cross-checked across sources · AI-generated, human-reviewed · Spotted a mistake? Tell us — help this leaf grow.
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