Multinomial Distribution -- How to calculate percentiles?

Based on your example, I am assuming that you have equal numbers of each marble.

Let $R,G,B$ be random variables representing the number of red, green, or blue marbles respectively that have been drawn. It sounds like you are looking for

$$P(R=r,G=g,B=b) = \dfrac{(r+g+b)!}{r!g!b!}\left(\dfrac{1}{3}\right)^{r+g+b}$$

Now, you can plot this as you vary $r,g,b$.

As for how many draws do you need on average to draw 25 of each, this is a modified coupon collector's problem. By linearity of expectation, drawing 25 of a single color marble will require $\dfrac{25}{\tfrac{1}{3}} = 75$ pulls of a marble. And drawing 25 of each would require $75+75+75 = 225$ draws on average.

Edit: this coupon collector solution does not work, unfortunately. The chances of getting multiples of the same thing are different from getting one of each of different things. I should have realized that. Through simulation, I am getting much smaller numbers. Like around $87.7$.

This gives a much closer estimate than my ill-conceived use of the Coupon Collector's Problem:

$$3\sum_{a=25}^{75}\sum_{b=25}^{75}(25+a+b)\dfrac{(24+a+b)!}{24!a!b!}\cdot \dfrac{1}{3^{25+a+b}} \approx 87.9057$$

Now, technically, I am undercounting, as $a,b$ should go to infinity, but I am also overcounting by multiplying by 3, as I am counting when $a=b=25$ three times, even though it should only be counted once. So, this is an estimate of the expected value.