black and white balls in the box A box contains $731$ black balls and $2000$ white balls. The following process is to be repeated as long as possible.
(1) arbitrarily select two balls from the box. If they are of the same color, throw them out and put a black ball into the box. (We have sufficient black balls for this).
(2) if they are of different colors, place the white ball back into the box and throw the black ball away.
What will happen at last? Will the process stop with a single black ball in the box or a single white ball in the box or with an empty box?
I am unable to decide how to start and in which direction? should we apply probability or what?

Let $B$ and $W$ denote the number of black and white balls respectively in the box. Note that the two operation affect the state of the system as:
\begin{align}
(B,W) \rightarrow (B-1,W) \\
(B,W) \rightarrow (B+1,W-2)
\end{align}
Every operation decreases the number of balls by 1. Considering the state transition diagram, if we start with an even number of white balls our final ball with be black, otherwise if we start with an odd number of white balls ($W$) the final ball will be white.
One can establish that by checking simple cases like $(3W,2B)$ and ($3W,3B$). With ($731B,2000W$), the final ball will be $B$.
