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A full joint distribution for the Toothache, Cavity and Catch is given in the table below.

  Toothache $\neg$ Toothache
Catch $\neg$ Catch Catch $\neg$ Catch
Cavity $0.108$ $0.012$ $0.072$ $0.008$
$\neg$ Cavity $0.016$ $0.064$ $0.144$ $0.576$

What is the probability of Cavity, given evidence of Toothache?

  1. $\langle 0.2, 0.8\rangle$
  2. $\langle 0.4, 0.8\rangle$
  3. $\langle 0.6, 0.8\rangle$
  4. $\langle 0.6, 0.4\rangle$
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$\underline{\textbf{Answer:}\Rightarrow}\;\left ( 0.6, 0.4 \right )$

$\underline{\textbf{Solution:}\Rightarrow}$

$\text{Probability of having cavity given the toothache} =\mathrm {P\left ( \dfrac{Cavity}{Toothache} \right ) \\= \dfrac{P\left ( Cavity \wedge Toothache \right )} {P \left (Toothache \right )}\\=\dfrac{\left ( 0.108 + 0.012\right )}{\left ( 0.108+0.012+0.016 +0.064 \right )} \\= \dfrac{0.12}{0.2} \\= 0.6}$

$\mathrm{P\left ( \dfrac{\displaystyle {\neg} Cavity}{Toothache} \right ) = \dfrac{\left ( 0.016 + 0.064\right )}{0.2} = 0.4}$

$\therefore \;\mathbf B$ is the right option.

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