Suppose you have $3$ jars with the following contents. Jar $1$ has $2$ white balls and $1$ black ball. Jar $2$ has $1$ white ball and $2$ black balls. Jar $3$ has $1$ white ball and $3$ black balls. One jar is to be selected, and then $1$ ball is to be drawn from the selected jar. The probabilities of selecting the first, second, and third jars are $\frac{1}{3},\frac{1}{2},$ and $\frac{1}{6}$ respectively. Find the probability the ball was drawn from Jar $1,$ given that the ball is white

What is the probability the ball was drawn from Jar $1,$ given that the ball is white?

$($Simplify your answer. Type an integer or a fraction.$)$