A box contains $2$ red marbles, $2$ green marbles, $2$ yellow marbles, $2$ blue marbles, and $2$ orange marbles. Carol takes $2$ marbles from the box at random, then Claudia takes $2$ of the remaining marbles at random, then Cheryl takes $2$, David takes $2$, and finally Emily takes the last $2$ marbles. What is the probability that Emily gets $2$ marbles of the same color?