Example 1
\(\text{Rolling a die: } P(\text{even}) = \frac{3}{6} = 0.5 \text{ or } 50\%\)
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Probability measures the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain). Several rules govern how probabilities combine for compound events:
- AND (Intersection): P(A ∩ B) = P(A) × P(B|A); for independent events, P(A) × P(B).
- OR (Union): P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
- Conditional: P(A|B) = P(A ∩ B) / P(B); the probability of A given that B occurred.
- Complement: P(A') = 1 − P(A); the probability that A does not occur.
- Independent Events: Two events are independent when P(A ∩ B) = P(A) × P(B); the occurrence of one does not affect the other.
- Mutually Exclusive: Two events are mutually exclusive when P(A ∩ B) = 0; they cannot occur at the same time.
Understanding these rules helps in tackling probability problems involving combinations of multiple events, conditional probabilities, and independence assumptions.
\(P(A \cup B) = P(A) + P(B) - P(A \cap B), P(A|B) = \frac{P(A \cap B)}{P(B)}, P(A') = 1 - P(A)\)
Worked Examples
Example 2
\(\text{Two coins: } P(\text{both heads}) = 0.5 \times 0.5 = 0.25 \text{ or } 25\%\)