There is a bin with 1 green ball, 1 red ball, 1 blue ball, and 1 yellow ball.
We draw 4 balls with replacement. What is the probability of drawing exactly 2 of a given color? Order doesn't matter, as long as we have exactly 2 balls drawn of a given color.
My attempt was as follows:
There are $\binom 4 2$ ways to select two "spots" in the 4 output spots. For each of those ways, the probability of having exactly two is $\frac 1 4 \cdot \frac 1 4 \cdot \frac 3 4 \cdot \frac 3 4$. Hence the final probability is just $\binom 4 2 \cdot \frac 1 4 \cdot \frac 1 4 \cdot \frac 3 4 \cdot \frac 3 4$.
Is this right?