Example 1
\(\text{Rolling a die: } P(\text{even}) = \frac{3}{6} = 0.5 \text{ or } 50\%\)
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Probability Calculations
Probability measures the likelihood of an event occurring, expressed as a number between 0 and 1 (or 0% and 100%). Key probability rules include:
- AND (Intersection): P(A and B) = P(A) × P(B) for independent events
- OR (Union): P(A or B) = P(A) + P(B) - P(A and B)
- Conditional: P(A|B) = P(A and B) / P(B) - probability of A given B
- Complement: P(A') = 1 - P(A) - probability of A not occurring
- Independent Events: P(A|B) = P(A) - occurrence of B doesn't affect A
- Mutually Exclusive: P(A and B) = 0 - events cannot occur together
Understanding these rules helps in calculating probabilities for complex scenarios involving multiple events, such as games of chance, risk assessment, and statistical inference.
\(P(A \cup B) = P(A) + P(B) - P(A \cap B), P(A|B) = \frac{P(A \cap B)}{P(B)}, P(A') = 1 - P(A)\)
Worked Examples
Example 2
\(\text{Two coins: } P(\text{both heads}) = 0.5 \times 0.5 = 0.25 \text{ or } 25\%\)