Exhaustive and Mutually Exclusive Events
Exhaustive and Mutually Exclusive Events
Exhaustive Events
A set of events is exhaustive if at least one of them must occur — together they cover every possible outcome. The probabilities of an exhaustive set sum to 1:
\[ P(A_1)+P(A_2)+\cdots+P(A_n)=1 \]Mutually Exclusive Events
Two events are mutually exclusive if they cannot both occur in the same trial — they share no outcomes. For mutually exclusive events:
\[ P(A \text{ or } B) = P(A) + P(B) \]This extends to any number of mutually exclusive events. On a Venn diagram, mutually exclusive events have no overlap.
Complementary Events
The complement of \( A \), written \( A' \) or \( \bar{A} \), is the event "A does not occur". \( A \) and \( A' \) are always mutually exclusive and exhaustive:
\[ P(A)+P(A')=1 \implies P(A')=1-P(A) \]The Addition Rule
For events that are not mutually exclusive (they can overlap):
\[ P(A \text{ or } B) = P(A)+P(B)-P(A \text{ and } B) \]Subtracting \( P(A\text{ and }B) \) corrects for counting the overlap twice.
Worked Examples
A bag has red, blue and green counters. \( P(R)=0.3 \), \( P(B)=0.45 \). Find \( P(G) \) and \( P(R\text{ or }B) \).
The events are exhaustive (only three colours): \( P(G)=1-0.3-0.45=0.25 \).
The events are mutually exclusive (a counter can only be one colour): \( P(R\text{ or }B)=0.3+0.45=0.75 \).
The probability of rain tomorrow is 0.35. Find the probability of no rain.
\[ P(\text{no rain})=1-0.35=0.65 \]In a group, \( P(\text{plays guitar})=0.4 \), \( P(\text{plays piano})=0.3 \), \( P(\text{plays both})=0.1 \). Find \( P(\text{plays at least one}) \).
\[ P(G\cup P)=0.4+0.3-0.1=0.6 \]Key Takeaways
- Exhaustive events: cover all outcomes; probabilities sum to 1.
- Mutually exclusive: cannot both occur; \( P(A\text{ or }B)=P(A)+P(B) \).
- Complement: \( P(A')=1-P(A) \).
- Non-mutually-exclusive: \( P(A\cup B)=P(A)+P(B)-P(A\cap B) \).
- Mutually exclusive events have no overlap on a Venn diagram.