Mutually Exclusive Events
Introduced| English |
|---|
| mutually exclusive/ˈmjuːtʃuːəli eksˈkluːsɪv/ |
| disjoint/dɪsˈdʒɔɪnt/ |
| addition rule/əˈdɪʃn ruːl/ |
| union/ˈjuːnɪən/ |
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: $P(A \cap B) = 0$.
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.
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.
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.
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.
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.
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.
Union of two events
The union is everything in A or B; subtract the overlap once.
P(A)=0.3, P(B)=0.4, P(A and B)=0.1. Find P(A or B).
0.3 + 0.4 − 0.1 = 0.6.
Two events are mutually exclusive. Then P(A and B) equals...
Disjoint events can't co-occur, so their intersection is 0.
For mutually exclusive events with P(A)=0.2 and P(B)=0.5, find P(A or B).
Disjoint → just add: 0.2 + 0.5 = 0.7.
For any two events, P(A or B) = P(A) + P(B), no subtraction needed.
You must subtract the overlap unless the events are disjoint.
Events that cannot happen at the same time are called mutually ___.
Mutually exclusive = disjoint = can't co-occur.