1.9 Properties of Probability¶
Key Properties:¶
Union of Two Events (A ∪ B):¶
- Represents all outcomes in event A, event B, or both.
- Venn Diagram: Union is the combined area of two circles (A and B), including the overlap.
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Example: For movies KGF (event A) and Kantara (event B):
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Interpretation: 76% chance either or both movies are a hit.
Intersection of Two Events (A ∩ B):¶
- Represents outcomes common to both events A and B.
- Venn Diagram: Intersection is the overlapping area of two circles (A and B).
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Example:
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Outcomes: {8, 9, 11, 12} with probabilities \( 0.08 + 0.02 + 0.06 + 0.02 \).
Addition Law:¶
- Formula: [ P(A ∪ B) = P(A) + P(B) - P(A ∩ B) ]
- Simplifies to \( P(A ∪ B) = P(A) + P(B) \) if A and B are mutually exclusive (no overlap).
Mutually Exclusive Events:¶
- Two events are mutually exclusive if they cannot occur simultaneously.
- Example: Tossing a coin results in either heads (A) or tails (B) but not both.
Basic Requirements for Probabilities:¶
for any event E.
- The sum of probabilities of all sample space outcomes = 1.
Applications:¶
- Union: Probability that either or both events occur (e.g., a hit movie).
- Intersection: Probability that both events occur (e.g., both movies are hits).
- Mutual Exclusivity: Simplifies addition law when events have no overlap.
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