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Probability Part 2

Section: Mathematics

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Question : 4 of 100

Marks: +1, -0

If

A

B

are two events such that

P(A \cap B)=0.1

P(A \mid B)

P(B \mid A)

are the roots of the equation

12 x^2

-7 x+1=0

, then the value of

\; \frac{P(\overline{A} \cup \overline{B})}{P(\overline{A} \cap \overline{B})}

[22 Jan 2025 Shift 2]

$\; \frac{9}{4}$
$\; \frac{5}{3}$
$\; \frac{4}{3}$
$\; \frac{7}{4}$

Solution:

P(A \cap B)=0.1,\ P(A \mid B)

and

P(B \mid A)

are the roots of the equation

12 x^2-7 x+1=0

\; \Rightarrow P(A \mid B) P(B \mid A) = \; \frac{1}{12}

\; \Rightarrow \; \; \; \frac{P(A \cap B)}{P(B)} \times \; \frac{P(A \cap B)}{P(A)} = \; \frac{1}{12}

\; \Rightarrow P(A) P(B)=12(0.1)^2

=0.12

\; \text{Also, } \; \; P(A \mid B)+P(B \mid A) = \; \frac{7}{12}

\Rightarrow P(A \cap B) \left( \; \frac{1}{P(B)} + \; \frac{1}{P(A)} \right) = \; \frac{7}{12}

\Rightarrow P(A)+P(B) = \; \frac{7}{12} \times \; \frac{0.12}{0.1}

\Rightarrow P(A)+P(B)=0.7

\; \; \frac{P(\overline{A} \cup \overline{B})}{P(\overline{A} \cap \overline{B})} = \; \frac{P(\overline{A \cap B})}{P(\overline{A \cup B})}

= \; \; \frac{1-P(A \cap B)}{1-P(A \cup B)}

= \; \; \frac{1-0.1}{1-(0.7-0.1)} = \; \frac{0.9}{0.4} = \; \frac{9}{4}

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