The problem states that P(A B) and P(B A) are the roots of the quadratic equation 12x²-7x+1=0. We can find these roots by factoring the quadratic equation or using the quadratic formula.

From the definition of conditional probability, we know that P(A B) = (P(A B))/(P(B)) and P(B A) = (P(A B))/(P(A)). We are given P(A B) = 0.1. Using the roots found, we can set up equations to find P(A) and P(B).

From the definition of conditional probability, we know that P(A B) = (P(A B))/(P(B)) and P(B A) = (P(A B))/(P(A)). We are given P(A B) = 0.1. Using the roots found, we can set up equations to find P(A) and P(B).

Using the conditional probability formulas and the given P(A B), we can calculate the individual probabilities P(A) and P(B).

Using De Morgan's laws, we know that P( A B) = P( A B) and P( A B) = P( A B). We can then use the complement rule P( E) = 1 - P(E) to find these values.

If $A$ and $B$ are two events such that $P(A \cap B)=0.1$ , and $P(A \mid B)$ and $P(B \mid A)$ are the roots of the equation $12x^2-7x+1=0$ , then the value of $\frac{P(\overline{A} \cup \overline{B})}{P(\overline{A} \cap \overline{B})}$ is:

Answer: $2.25$

Step-by-step solution

Step 1: Find the roots of the quadratic equation

The problem states that $P(A \mid B)$ and $P(B \mid A)$ are the roots of the quadratic equation $12x^2-7x+1=0$ . We can find these roots by factoring the quadratic equation or using the quadratic formula.

Step 2: Identify conditional probabilities

From the definition of conditional probability, we know that $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$ and $P(B \mid A) = \frac{P(A \cap B)}{P(A)}$ . We are given $P(A \cap B) = 0.1$ . Using the roots found, we can set up equations to find $P(A)$ and $P(B)$ .

Step 3: Calculate $P(A)$ and $P(B)$

Using the conditional probability formulas and the given $P(A \cap B)$ , we can calculate the individual probabilities $P(A)$ and $P(B)$ .

Step 4: Calculate $P(A \cup B)$

We use the formula for the probability of the union of two events to find $P(A \cup B)$ .

Step 5: Calculate $P(\overline{A} \cup \overline{B})$ and $P(\overline{A} \cap \overline{B})$

Using De Morgan's laws, we know that $P(\overline{A} \cup \overline{B}) = P(\overline{A \cap B})$ and $P(\overline{A} \cap \overline{B}) = P(\overline{A \cup B})$ . We can then use the complement rule $P(\overline{E}) = 1 - P(E)$ to find these values.

Step 6: Calculate the final ratio

Finally, we substitute the calculated values into the required ratio to find the answer.

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