Editing Bayes' theorem (section)

====Example====
[[File:Bayes theorem simple example tree.svg|thumb|Tree diagram illustrating the beetle example. ''R, C, P'' and <math> \overline{P} </math> are the events rare, common, pattern and no pattern. Percentages in parentheses are calculated. Three independent values are given, so it is possible to calculate the inverse tree.]]

An [[Entomology|entomologist]] spots what might, due to the pattern on its back, be a rare [[subspecies]] of [[beetle]]. A full 98% of the members of the rare subspecies have the pattern, so ''P''(Pattern&nbsp;|&nbsp;Rare) = 98%. Only 5% of members of the common subspecies have the pattern. The rare subspecies is 0.1% of the total population. How likely is the beetle having the pattern to be rare: what is ''P''(Rare&nbsp;|&nbsp;Pattern)?

From the extended form of Bayes' theorem (since any beetle is either rare or common),

: <math display=block>
\begin{align}
P(\text{Rare} \vert \text{Pattern}) &= \frac{P(\text{Pattern} \vert \text{Rare})\,P(\text{Rare})} {P(\text{Pattern})}\\
[8pt] &= \tfrac{P(\text{Pattern}\vert \text{Rare})\,P(\text{Rare})} {P(\text{Pattern} \vert \text{Rare})\,P(\text{Rare}) + P(\text{Pattern}\vert \text{Common})\,P(\text{Common})}\\
[8pt] &= \frac{0.98 \times 0.001} {0.98 \times 0.001 + 0.05 \times 0.999}\\
[8pt] &\approx 1.9\%
\end{align}
</math>