A central theorem in statistics is Bayes' theorem. This theorem plays a similar role as the good old Pythagoras' theorem in geometry. Bayes' theorem is extremely simple to derive. But to do so we need some basic axioms from statistics.
Assume we have two domains of events \( X=[x_0,x_1,\dots,x_{n-1}] \) and \( Y=[y_0,y_1,\dots,y_{n-1}] \).
We define also the likelihood for \( X \) and \( Y \) as \( p(X) \) and \( p(Y) \) respectively. The likelihood of a specific event \( x_i \) (or \( y_i \)) is then written as \( p(X=x_i) \) or just \( p(x_i)=p_i \).
where we read \( p(X\vert Y) \) as the likelihood of obtaining \( X \) given \( Y \).
If we have independent events then \( p(X,Y)=p(X)p(Y) \).