Mutually Exclusive Events · 互斥事件
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| mutually exclusive/ˈmjuːtʃuːəli eksˈkluːsɪv/ | 互斥 | hù chì |
| disjoint/dɪsˈdʒɔɪnt/ | 不相交 | bù xiāng jiāo |
| addition rule/əˈdɪʃn ruːl/ | 加法法则 | jiā fǎ fǎ zé |
| union/ˈjuːnɪən/ | 并集 | bìng jí |
Events that can't overlap
- Two events are mutually exclusive 互斥 (or disjoint 不相交) if they can't happen together.
- Rolling a die: "get a $2$" and "get a $5$" can't both occur on one roll.
- On a Venn diagram, disjoint events are circles that don't overlap.
- Their intersection is empty: $P(A \cap B) = 0$.
不能重叠的事件
- 如果两个事件不能同时发生，它们就是互斥（或不相交）的。
- 掷骰子：“得到 $2$”和“得到 $5$”不能在一次投掷中同时发生。
- 在维恩图上，不相交的事件是不重叠的圆。
- 它们的交集为空：$P(A \cap B) = 0$。
The general addition rule
- To find $P(A \text{ or } B)$ — the union 并集 — use the addition rule 加法法则:
-
$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
- You subtract the overlap so it isn't counted twice.
- This rule works for any two events, overlapping or not.
一般加法法则
- 要求 $P(A \text{ or } B)$——即并集——用加法法则：
-
$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
- 你要减去重叠部分，以免它被重复计数。
- 这个法则对任意两个事件都成立，无论是否重叠。
The disjoint shortcut
- If $A$ and $B$ are mutually exclusive, their overlap is $0$.
- So the rule simplifies to $P(A \cup B) = P(A) + P(B)$.
- Just add — there's nothing to subtract.
- This shortcut is only valid when the events truly can't co-occur.
互斥的捷径
- 如果 $A$ 和 $B$ 互斥，它们的重叠为 $0$。
- 所以法则简化为 $P(A \cup B) = P(A) + P(B)$。
- 直接相加——没有什么可减的。
- 这个捷径只在事件确实不能共存时才有效。
From a table or diagram
- A two-way table or Venn diagram gives you the pieces directly.
- Read off $P(A)$, $P(B)$, and the overlap $P(A \cap B)$, then apply the rule.
- The overlap is the cell (or region) belonging to both events.
- Careful bookkeeping of the overlap is the whole trick.
从表格或图
- 双向表或维恩图直接给你各个部分。
- 读出 $P(A)$、$P(B)$ 和重叠 $P(A \cap B)$，再套用法则。
- 重叠是同时属于两个事件的那个单元格（或区域）。
- 仔细核对重叠部分就是全部的诀窍。
Don't forget to subtract the overlap. The general rule is $P(A\cup B)=P(A)+P(B)-P(A\cap B)$; dropping the $-P(A\cap B)$ double-counts the overlap and overstates the probability. Only when events are mutually exclusive (overlap $=0$) may you just add. Check for overlap before choosing the shortcut.
别忘了减去重叠。一般法则是 $P(A\cup B)=P(A)+P(B)-P(A\cap B)$；丢掉 $-P(A\cap B)$ 会重复计数重叠部分，从而高估概率。只有当事件互斥（重叠 $=0$）时，你才可以直接相加。选择捷径之前先检查是否有重叠。
Draw one card. $P(\text{king}) = 4/52$, $P(\text{heart}) = 13/52$, $P(\text{king and heart}) = 1/52$.
- Union: $P(\text{king or heart}) = \tfrac{4}{52} + \tfrac{13}{52} - \tfrac{1}{52} = \tfrac{16}{52}$.
- The $-\tfrac{1}{52}$ removes the king of hearts, counted in both.
- "King" and "queen," by contrast, are disjoint → just add.
抽一张牌。$P(\text{king}) = 4/52$，$P(\text{heart}) = 13/52$，$P(\text{king and heart}) = 1/52$。
- 并集：$P(\text{king or heart}) = \tfrac{4}{52} + \tfrac{13}{52} - \tfrac{1}{52} = \tfrac{16}{52}$。
- 那个 $-\tfrac{1}{52}$ 去掉了红心 K，它在两者中都被数了。
- 相比之下，“K”和“Q”是不相交的 → 直接相加。
Events are mutually exclusive (disjoint) when they can't co-occur, so $P(A \cap B) = 0$. The addition rule $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ finds a union; for disjoint events it simplifies to $P(A) + P(B)$. Always subtract the overlap unless it's truly zero.
当事件不能共存时它们互斥（不相交），此时 $P(A \cap B) = 0$。加法法则 $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ 求并集；对不相交的事件它简化为 $P(A) + P(B)$。除非重叠确实为零，否则总要减去重叠。
Union of two events · 两个事件的并集
The union is everything in A or B; subtract the overlap once. · 并集是 A 或 B 中的一切；重叠只减一次。
P(A)=0.3, P(B)=0.4, P(A and B)=0.1. Find P(A or B). · P(A)=0.3，P(B)=0.4，P(A 且 B)=0.1。求 P(A 或 B)。
0.3 + 0.4 − 0.1 = 0.6. · 0.3 + 0.4 − 0.1 = 0.6。
Two events are mutually exclusive. Then P(A and B) equals... · 两个事件互斥。那么 P(A 且 B) 等于……
Disjoint events can't co-occur, so their intersection is 0. · 不相交事件不能共存，所以交集为 0。
For mutually exclusive events with P(A)=0.2 and P(B)=0.5, find P(A or B). · 对互斥事件 P(A)=0.2、P(B)=0.5，求 P(A 或 B)。
Disjoint → just add: 0.2 + 0.5 = 0.7. · 不相交 → 直接相加：0.2 + 0.5 = 0.7。
For any two events, P(A or B) = P(A) + P(B), no subtraction needed. · 对任意两个事件，P(A 或 B) = P(A) + P(B)，不需要减法。
You must subtract the overlap unless the events are disjoint. · 除非事件不相交，否则必须减去重叠。
Events that cannot happen at the same time are called mutually ___. · 不能同时发生的事件叫做互相 ___（填英文一词，即“互斥”对应词）。
Mutually exclusive = disjoint = can't co-occur. · 互斥 = 不相交 = 不能共存。