Independent Events · 独立事件
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| independent events/ˌɪndɪˈpendənt ɪˈvents/ | 独立事件 | dú lì shì jiàn |
When knowing one tells you nothing
- Two events are independent 独立事件 if one happening doesn't change the other's probability.
- Formally: $P(A \mid B) = P(A)$ — the condition makes no difference.
- Coin tosses are independent: the first result tells you nothing about the second.
- If knowing $B$ does shift $P(A)$, the events are dependent.
当知道一个却毫无用处
- 如果一个事件的发生不改变另一个的概率，这两个事件就是独立事件。
- 形式上：$P(A \mid B) = P(A)$——那个条件毫无影响。
- 抛硬币是独立的：第一次的结果对第二次毫无预示。
- 如果知道 $B$ 确实改变了 $P(A)$，这两个事件就是相依的。
The multiplication shortcut
- For independent events, the multiplication rule simplifies:
-
$$P(A \cap B) = P(A)\,P(B)$$
- Since $P(A \mid B) = P(A)$, the condition drops out — just multiply.
- This "and = multiply" only holds when events are independent.
乘法捷径
- 对独立事件，乘法法则可以简化：
-
$$P(A \cap B) = P(A)\,P(B)$$
- 因为 $P(A \mid B) = P(A)$，条件就消掉了——直接相乘。
- 这个“且 = 相乘”只在事件独立时才成立。
Independent ≠ mutually exclusive
- These two ideas are opposites, not synonyms — a classic mix-up.
- Mutually exclusive: they can't both happen ($P(A \cap B) = 0$).
- Independent: they can both happen, and one doesn't affect the other.
- In fact, two events with nonzero probability that are disjoint are always dependent.
独立 ≠ 互斥
- 这两个概念是相反的，不是同义词——一个经典的混淆。
- **互斥：**它们不能同时发生（$P(A \cap B) = 0$）。
- **独立：**它们可以同时发生，且一个不影响另一个。
- 事实上，两个概率非零且不相交的事件总是相依的。
Combining rules
- Compound problems chain the addition and multiplication rules.
- "At least one" is often easiest via the complement: $1 - P(\text{none})$.
- $P(\text{none}) = P(\text{not }A)\,P(\text{not }B)\cdots$ for independent events.
- Break the event into "and"s (multiply) and "or"s (add), step by step.
组合法则
- 复合问题把加法和乘法法则串起来。
- “至少一个”常常用补集最简单：$1 - P(\text{none})$。
- 对独立事件，$P(\text{none}) = P(\text{not }A)\,P(\text{not }B)\cdots$。
- 把事件拆成“且”（相乘）和“或”（相加），一步步来。
Independent and mutually exclusive are not the same — they're opposites. Disjoint events can't co-occur, so learning one happened tells you the other didn't — that's maximal dependence. Only use $P(A\cap B)=P(A)P(B)$ after you've checked independence; never assume it just because the problem gives you $P(A)$ and $P(B)$.
独立和互斥不是一回事——它们是相反的。不相交的事件不能共存，所以得知其中一个发生了就说明另一个没发生——这是最大程度的相依。只有在你检查过独立性之后才使用 $P(A\cap B)=P(A)P(B)$；绝不要仅因为题目给了 $P(A)$ 和 $P(B)$ 就假定它。
Flip a fair coin twice (independent flips).
- $P(\text{two heads}) = P(H)\,P(H) = 0.5 \times 0.5 = 0.25$.
- $P(\text{at least one head}) = 1 - P(\text{no heads}) = 1 - 0.5 \times 0.5 = 0.75$.
- The complement turns a messy "or" into a clean "and."
抛一枚均匀硬币两次（独立抛掷）。
- $P(\text{two heads}) = P(H)\,P(H) = 0.5 \times 0.5 = 0.25$。
- $P(\text{at least one head}) = 1 - P(\text{no heads}) = 1 - 0.5 \times 0.5 = 0.75$。
- 补集把一个麻烦的“或”变成一个干净的“且”。
Events are independent when $P(A \mid B) = P(A)$; then $P(A \cap B) = P(A)\,P(B)$ (multiply). This is not the same as mutually exclusive (which means $P(A\cap B)=0$) — they're opposites. Combine the addition and multiplication rules for compound events, often using the complement for "at least one."
当 $P(A \mid B) = P(A)$ 时事件独立；此时 $P(A \cap B) = P(A)\,P(B)$（相乘）。这与互斥（意味着 $P(A\cap B)=0$）不同——它们是相反的。对复合事件组合使用加法和乘法法则，“至少一个”常用补集。
Independent events branch by branch · 独立事件的逐枝分解
For independent events, multiply along the branches. · 对独立事件，沿着树枝相乘。
Two independent events have P(A)=0.5 and P(B)=0.5. Find P(A and B). · 两个独立事件 P(A)=0.5、P(B)=0.5。求 P(A 且 B)。
Independent → multiply: 0.5 × 0.5 = 0.25. · 独立 → 相乘：0.5 × 0.5 = 0.25。
Independent events and mutually exclusive events are the same thing. · 独立事件和互斥事件是同一回事。
They're opposites: disjoint events are actually dependent. · 它们相反：不相交事件其实是相依的。
Flip a fair coin twice. Find P(at least one head). · 抛一枚均匀硬币两次。求 P(至少一个正面)。
1 − P(no heads) = 1 − 0.5×0.5 = 0.75. · 1 − P(没有正面) = 1 − 0.5×0.5 = 0.75。
Events A and B are independent means... · 事件 A 和 B 独立意味着……
Independence: the condition doesn't change A's probability. · 独立：条件不改变 A 的概率。
For 'at least one', it is often easiest to use the ___ rule (1 minus P(none)). · 对“至少一个”，常常最容易用 ___ 法则（1 减 P(一个都没有)）（填英文一词）。
P(at least one) = 1 − P(none). · P(至少一个) = 1 − P(一个都没有)。