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 dfrac P(overline A cup overline B ) P(overline A cap overline B ) is:

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EEJ MAIN Mathematics QUESTION #966

Question 1

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 \(\dfrac{P(\overline{A} \cup \overline{B})}{P(\overline{A} \cap \overline{B})}\) is:

\(\dfrac{4}{3}\)
\(\dfrac{7}{4}\)
\(\dfrac{5}{3}\)
\(\dfrac{9}{4}\)✔️

Correct Answer Explanation

Roots of \(12x^2-7x+1=0\) are \(\frac{1}{3}\) and \(\frac{1}{4}\). Since \(P(A|B)=\frac{P(A\cap B)}{P(B)}=\frac{1}{3}\) gives \(P(B)=0.3\), and \(P(B|A)=\frac{1}{4}\) gives \(P(A)=0.4\). Then \(P(A\cup B)=0.6\), \(P(\overline{A}\cup\overline{B})=1-0.1=0.9\), \(P(\overline{A}\cap\overline{B})=0.4\). Ratio \(=\frac{0.9}{0.4}=\frac{9}{4}\).

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