[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Suppose that [math]A[/math] and [math]B[/math] are events such that [math]P(A|B) = P(B|A)[/math] and [math]P(A \cup B) = 1[/math] and [math]P(A \cap B) \gt 0[/math]. Prove that [math]P(A) \gt 1/2[/math].