Probability question with balls in box

Change question title if you can come up with better one

In box we have $k$ white, $m$ blue and $n$ red balls. From the box one after another we fetch all balls and we write down color of that ball. What's the probability that in our list of colors, white color will appear before a blue.

So, as I understood we can have any amount of red balls in a row and then white must appear.

For example: W, RW, RRW, RRRW, R..........W

So I need to calculate something like that:
$$\frac{k}{k+m+n}+\frac{n}{k+m+n}\frac{k}{k+m+n-1}+\frac{n}{k+m+n}\frac{n-1}{k+m+n-1}\frac{k}{k+m+n-2}+...$$

until all red balls appeared and finally white

But I don't know what appears here and how to write it. Or maybe there's even more easier way to solve it?

Answer

The number of reds is just as irrelevant as whether the sun is shining or not. The probability is $\dfrac{k}{k+m}$.

Remark: Let $t$ be the total number of balls. So $t=k+m+n$. Then, somewhat as you wrote, but with a small correction, the probability that a white comes before a blue is
$$\frac{k}{t}+\frac{n}{t}\frac{k}{t-1}+\frac{n}{t}\frac{n-1}{t-1}\frac{k}{t-2}+\cdots.$$
As was explained earlier, this sum is obviously $\frac{k}{k+m}$. So by finding the probability in two different ways, we obtain a potentially useful combinatorial identity.