There are two bags, $A$ and $B$ having $2$ white balls and $1$ black ball in Bag $A$ and $1$ white ball and $3$ black balls in bag $B$. A bag is chosen at random and two balls are drawn. The balls were $1$ white and $1$ black ball. What is the probability that is from bag $B$?
I have solved it visually using a tree structure, but how I express it in mathematical expression.