In an urn there are $a$ azure balls and $c$ carmine balls, $ac\ne0$. To begin with, you randomly pick a ball, throw it away, and then each time you randomly pick a ball, if it has the same color with its predecessor, throw it away, otherwise put it back. Then what's the probability that the last one thrown from the urn is azure?
For instance, a possible round:
draw urn
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AAACCCCC
A AACCCCC
C AACCCCC
C AACCCC
C AACCC
A AACCC
C AACCC
A AACCC
C AACCC
C AACC
C AAC
C AA
A AA
A A
A -
In this round, the last one thrown is an azure ball.