If A and B are two events such that P(A∩B)=0.1, and P(A∣B) and P(B∣A) are the roots of the equation 12x2−7x+1=0, then the value of P(A∩B)P(A∪B) is:
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Step-by-Step Solution
Step 1: Find the roots of the quadratic equation
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.
Step 2: Identify conditional probabilities
From the definition of conditional probability, we know that P(A∣B)=P(B)P(A∩B) and P(B∣A)=P(A)P(A∩B). We are given P(A∩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∩B), we can calculate the individual probabilities P(A) and P(B).
Step 4: Calculate P(A∪B)
We use the formula for the probability of the union of two events to find P(A∪B).
Step 5: Calculate P(A∪B) and P(A∩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.
Step 6: Calculate the final ratio
Finally, we substitute the calculated values into the required ratio to find the answer.