Key Concepts and Formulas

1. Conditional Probability: The probability of an event $E_1$ occurring given that another event $E_2$ has already occurred is defined as: $P(E_1 \mid E_2) = \frac{P(E_1 \cap E_2)}{P(E_2)}, \quad \text{provided } P(E_2) > 0$
2. Probability of the Union of Two Events: For any two events $A$ and $B$, the probability that at least one of them occurs is: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
3. Complement Rule & De Morgan's Laws:
   • The probability of the complement of an event $E$ is $P(\bar{E}) = 1 - P(E)$.
   • De Morgan's Laws for probabilities are:
     • $P(\bar{A} \cup \bar{B}) = P(\overline{A \cap B})$ (The complement of an intersection is the union of complements).
     • $P(\bar{A} \cap \bar{B}) = P(\overline{A \cup B})$ (The complement of a union is the intersection of complements).

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Step-by-Step Solution

Step 1: Determine the values of the conditional probabilities.

• What we are doing: The problem states that $P(A \mid B)$ and $P(B \mid A)$ are the roots of the quadratic equation $12x^2 - 7x + 1 = 0$. Our initial task is to solve this equation to find these numerical values.
• Why this is important: These conditional probabilities are critical pieces of information. They act as a bridge, allowing us to eventually determine the individual probabilities $P(A)$ and $P(B)$, which are essential for subsequent calculations.
• Mathematical Working: We solve the quadratic equation $12x^2 - 7x + 1 = 0$ by factoring. We look for two numbers that multiply to $12 \times 1 = 12$ and add up to $-7$. These numbers are $-3$ and $-4$. $12x^2 - 4x - 3x + 1 = 0$ Factor by grouping: $4x(3x - 1) - 1(3x - 1) = 0$ $(4x - 1)(3x - 1) = 0$ Setting each factor to zero gives us the roots: $4x - 1 = 0 \implies x = \frac{1}{4}$ $3x - 1 = 0 \implies x = \frac{1}{3}$ Thus, the roots of the equation are $\frac{1}{3}$ and $\frac{1}{4}$. We assign these to the conditional probabilities. The specific assignment (which is $P(A \mid B)$ and which is $P(B \mid A)$) does not affect the final answer due to the symmetric nature of how $P(A)$ and $P(B)$ will be derived. Let $P(A \mid B) = \frac{1}{3}$ and $P(B \mid A) = \frac{1}{4}$.
• Reasoning: This step directly translates the given information into concrete numerical values, forming the foundation for the rest of the problem.

Step 2: Calculate $P(A)$ and $P(B)$ using the definition of conditional probability.

• What we are doing: We will now use the definition of conditional probability, along with the given $P(A \cap B) = 0.1$ and the conditional probabilities found in Step 1, to determine the individual probabilities $P(A)$ and $P(B)$.
• Why this is important: Knowing $P(A)$ and $P(B)$ is crucial for calculating the probability of their union, $P(A \cup B)$, which is a necessary intermediate step for evaluating the target ratio.
• Mathematical Working: Using the definition $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$: $\frac{1}{3} = \frac{0.1}{P(B)}$ Solving for $P(B)$: $P(B) = 3 \times 0.1 \implies P(B) = 0.3$ Using the definition $P(B \mid A) = \frac{P(A \cap B)}{P(A)}$: $\frac{1}{4} = \frac{0.1}{P(A)}$ Solving for $P(A)$: $P(A) = 4 \times 0.1 \implies P(A) = 0.4$
• Reasoning: By rearranging the conditional probability formula, we effectively isolate and solve for the unknown individual probabilities $P(A)$ and $P(B)$. Both $0.3$ and $0.4$ are valid probabilities (between 0 and 1).

Step 3: Calculate the probability of the union $P(A \cup B)$.

• What we are doing: Now that we have $P(A)$, $P(B)$, and the given $P(A \cap B)$, we can calculate the probability that at least one of the events $A$ or $B$ occurs using the formula for the probability of the union of two events.
• Why this is important: The denominator of the expression we ultimately need to evaluate is $P(\bar{A} \cap \bar{B})$. According to De Morgan's Laws, this is equivalent to $P(\overline{A \cup B})$, which can be found using the complement rule as $1 - P(A \cup B)$. Therefore, calculating $P(A \cup B)$ is a necessary step towards finding the denominator.
• Mathematical Working: Using the formula $P(A \cup B) = P(A) + P(B) - P(A \cap B)$: $P(A \cup B) = 0.4 + 0.3 - 0.1$ $P(A \cup B) = 0.7 - 0.1$ $P(A \cup B) = 0.6$
• Reasoning: This is a direct application of the union formula, providing a key intermediate value that will be used in the subsequent steps to simplify the target expression.

Step 4: Calculate the numerator $P(\bar{A} \cup \bar{B})$.

• What we are doing: We will simplify and calculate the numerator of the target ratio, $P(\bar{A} \cup \bar{B})$, by applying De Morgan's Laws and the complement rule.
• Why this is important: This step directly transforms a seemingly complex expression involving complements of individual events into a simpler form that can be calculated using the given probability of the intersection, $P(A \cap B)$.
• Mathematical Working: Using De Morgan's Law, the union of complements is the complement of the intersection: $P(\bar{A} \cup \bar{B}) = P(\overline{A \cap B})$ Now, using the complement rule: $P(\overline{A \cap B}) = 1 - P(A \cap B)$ We are given $P(A \cap B) = 0.1$. $P(\bar{A} \cup \bar{B}) = 1 - 0.1 = 0.9$
• Reasoning: De Morgan's Law provides an elegant way to convert the union of complements into the complement of an intersection, which can then be easily calculated using the basic complement rule and the given information.

Step 5: Calculate the denominator $P(\bar{A} \cap \bar{B})$.

• What we are doing: We will simplify and calculate the denominator of the target ratio, $P(\bar{A} \cap \bar{B})$, by applying De Morgan's Laws and the complement rule.
• Why this is important: Similar to the numerator, this step is crucial for transforming the denominator into a calculable form using the $P(A \cup B)$ value derived in Step 3.
• Mathematical Working: Using De Morgan's Law, the intersection of complements is the complement of the union: $P(\bar{A} \cap \bar{B}) = P(\overline{A \cup B})$ Now, using the complement rule: $P(\overline{A \cup B}) = 1 - P(A \cup B)$ We calculated $P(A \cup B) = 0.6$ in Step 3. $P(\bar{A} \cap \bar{B}) = 1 - 0.6 = 0.4$
• Reasoning: De Morgan's Law allows us to express the intersection of complements as the complement of the union, which can then be directly calculated using the complement rule and our previously determined $P(A \cup B)$.

Step 6: Calculate the final ratio.

• What we are doing: With both the numerator and the denominator calculated, the final step is to perform the division to obtain the answer requested by the question.
• Why this is important: This is the concluding step that synthesizes all the intermediate results to arrive at the final solution.
• Mathematical Working: We have the numerator $P(\bar{A} \cup \bar{B}) = 0.9$ and the denominator $P(\bar{A} \cap \bar{B}) = 0.4$. $\frac{P(\bar{A} \cup \bar{B})}{P(\bar{A} \cap \bar{B})} = \frac{0.9}{0.4}$ To simplify the fraction, we can multiply both the numerator and the denominator by 10: $\frac{0.9 \times 10}{0.4 \times 10} = \frac{9}{4}$
• Reasoning: The final ratio is obtained by straightforward division of the two components derived in the preceding steps.

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Common Mistakes & Tips

• Distinguishing Conditional Probability: A common error is to confuse $P(A \mid B)$ with $P(A \cap B)$. Remember that $P(A \mid B)$ is the probability of $A$ given that $B$ has already occurred, while $P(A \cap B)$ is the probability that both $A$ and $B$ occur. Their formulas are distinct.
• Accurate De Morgan's Law Application: Ensure you correctly apply De Morgan's Laws: $P(\bar{A} \cup \bar{B}) = P(\overline{A \cap B})$ and $P(\bar{A} \cap \bar{B}) = P(\overline{A \cup B})$. A frequent mistake is to incorrectly swap these relationships.
• Probability Range Check: Always perform a quick check to ensure that any probability value you calculate (e.g., $P(A)$, $P(B)$, $P(A \cup B)$) lies within the valid range of $[0, 1]$. This can help in catching calculation errors early.

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Summary

This problem effectively tests several fundamental concepts in probability. We began by solving a quadratic equation to find the values of the conditional probabilities $P(A \mid B)$ and $P(B \mid A)$. Using these values along with the given $P(A \cap B)$, we then calculated the individual probabilities $P(A)$ and $P(B)$. Subsequently, we determined the probability of the union, $P(A \cup B)$. Finally, by expertly applying De Morgan's Laws and the complement rule, we transformed the numerator $P(\bar{A} \cup \bar{B})$ into $1 - P(A \cap B)$ and the denominator $P(\bar{A} \cap \bar{B})$ into $1 - P(A \cup B)$, which allowed for a straightforward calculation of the required ratio.

The final answer is $\boxed{\frac{9}{4}}$. This corresponds to option (C).