If A and B are two events such that \( P(A \cap B) = 0.1 \), and \( P(A|B) \) and \( P(B|A) \) are the roots of the equation \( 12x^2 - 7x + 1 = 0 \), then the value of \( \frac{P(A \cup B)}{P(A \cap B)} \) is:
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For problems involving conditional probabilities:
- Recall that \( P(A|B) = \frac{P(A \cap B)}{P(B)} \) and \( P(B|A) = \frac{P(A \cap B)}{P(A)} \).
- Use these relationships to form equations based on the given conditions, and then solve for the required ratio of probabilities.
Given that \( P(A|B) \) and \( P(B|A) \) are the roots of \( 12x^2 - 7x + 1 = 0 \), Vieta's formulas provide the sum and product of these roots:\[P(A|B) + P(B|A) = \frac{7}{12}, \quad P(A|B) \cdot P(B|A) = \frac{1}{12}.\]Applying conditional probability relationships allows calculation of \( \frac{P(A \cup B)}{P(A \cap B)} \), which results in \( \frac{4}{3} \).
