Question #41716

Let $A$ and $B$ be two disjoint events in a sample space such that $P(A) = 3P(B)$. Consider three points in the Cartesian plane: $V_1 = (P(A), P(B))$, $V_2 = (P(A \cup B), P(A \cap B))$, and $V_3 = (P(A^c), P(B^c))$. Let $M$ be the $3 \times 3$ matrix whose rows are $(x_i, y_i, 1)$ for $i=1,2,3$, where $(x_i, y_i)$ are the coordinates of $V_i$ in the standard $(X, Y)$ coordinate system. Let $D = \det(M)$. A new coordinate system $(X', Y')$ is defined by the transformation $X' = X - P(A)$ and $Y' = P(B) - Y$. Let $M'$ be the corresponding matrix using the coordinates of the same points in the $(X', Y')$ system, and let $D' = \det(M')$. If $D - D' = \frac{1}{4}$, what is the value of $P(A \cup B)$?