Probability
Probability
Allen Star Batch
Grade 12
Question:
If $A$ and $B$ are two events such that $P(A) = 0.3, P(B) = 0.25, P\left(A \cap B\right) = 0.2$, then $10P\left(\frac{A^C}{B^C}\right)$ is equal to ______.
Step-by-Step Solution
Key Concept: Apply conditional probability formula P(A|B) = P(A∩B)/P(B) with complements: P(A^c|B^c) = P(A^c∩B^c)/P(B^c), where P(A^c∩B^c) = 1 - P(A∪B) using De Morgan's law and the union formula P(A∪B) = P(A) + P(B) - P(A∩B).
Using the complement and conditional probability: $P\left(\frac{A^c}{B^c}\right) = 1 - P\left(\frac{A}{B^c}\right) = 1 - \frac{P(A^c \cap B^c)}{P(B^c)} = 1 - \frac{1 - P(A \cup B)}{1 - P(B)} = 1 - \frac{1 - P(A) - P(B) + P(A \cap B)}{1 - P(B)} = 1 - \frac{0.65}{0.75} = \frac{2}{15}$.
Correct Answer: 1.33