Normalisation (probabilities)

We can use $\alpha =1/P(\text{toothache})$ (calculation not necessary) to normalise the following:

$$\begin{array}{rl}P(\text{Cavity}|\text{toothache})& =\alpha P(\text{Cavity},\text{toothache})\\ & =\alpha [P(\text{Cavity},\text{toothache},\text{catch}),P(\text{Cavity},\text{toothache},\neg \text{catch})]\\ & =\alpha \left[\begin{array}{c}\left(\begin{array}{c}0.108\\ 0.016\end{array}\right)+\left(\begin{array}{c}0.012\\ 0.064\end{array}\right)\end{array}\right]\\ & =\left(\begin{array}{c}0.12\\ 0.08\end{array}\right)\\ & =\left(\begin{array}{c}0.6\\ 0.4\end{array}\right)\end{array}$$

The source of these numbers is from the follow joint distribution table:

	$toothache$		$\neg toothache$
	$catch$	$\neg catch$	$catch$	$\neg catch$
$cavity$	$0.108$	$0.012$	$0.072$	$0.008$
$\neg cavity$	$0.016$	$0.064$	$0.144$	$0.576$