In a box there are $R$ red balls and $B$ blue balls. $R$ and $B$ are even numbers, and $R>B$.
Initially a random ball is drawn from the box, we take a look on its color and keep the ball out of the box. After, we keep drawning balls from the box until we get a ball where its color is different from the first one. When it happens, we place this ball(the one with different color) back in the box and the process is restarted until there is no balls left.
The question is, what is the probability of the last ball picked from the ball is blue?
What I tried to do was, solve the problem backwards, that means that for the blue ball be the last one the drawns in the box must be specific. But I got nowhere with this idea.
The other thing that I tried was to write all possibilities with $R=4$ and $B=2$, but I don't know how to deal with the restart of the process.
Someone has any ideia how to model this problem?
Here is an example:
Let's say that we have $50$ red balls and $14$ blue balls.
Suppose that the firts drawn ball is red, in a such a way that now there are $49$ balls and $14$ blue balls inside the box. The second drawn is a red ball too, so now we have $48$ red balls and $14$ blue balls inside the box. Let's say that in the third drawn the ball is blue, as the last ball is a different color from the others taken out of the box, the blue ball is put back in the box and the process restarts with $48$ red balls and 14 blue balls. This keep going until there is no balls left.