How do I solve this probability problem of randomly drawing balls from a urn? 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
----------------
        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.
 A: As Chris wrote in a comment, the colour of the previously discarded ball must be included in the state. So denote the probability that the last of $a$ azure balls and $c$ carmine balls that gets discarded is azure by $A(a,c)$ if the previously discarded ball was azure, and by $C(a,c)$ if it was carmine. Then
$$A(a,c)=\frac a{a+c}A(a-1,c)+\frac c{a+c}C(a,c)$$
and
$$C(a,c)=\frac a{a+c}A(a,c)+\frac c{a+c}C(a,c-1)\;.$$
Substituting these equations into each other leads to the recurrences
$$A(a,c)=\frac a{a+c}A(a-1,c)+\frac c{a+c}\left(\frac a{a+c}A(a,c)+\frac c{a+c}C(a,c-1)\right)$$
and
$$C(a,c)=\frac a{a+c}\left(\frac a{a+c}A(a-1,c)+\frac c{a+c}C(a,c)\right)+\frac c{a+c}C(a,c-1)\;,$$
which simplify to
$$
(a^2+ac+c^2)A(a,c)=(a^2+ac)A(a-1,c)+c^2C(a,c-1)
$$
and
$$
(a^2+ac+c^2)C(a,c)=a^2A(a-1,c)+(ac+c^2)C(a,c-1)\;,
$$
respectively.
The initial conditions are $A(a,1)=1$, $C(1,c)=0$, $A(0,c)=0$ for $c\gt1$ and $C(a,0)=1$ for $a\gt1$.
I don't currently see how to solve this in closed form; I'll compute some values, check OEIS and think about asymptotics when I have more time later on.
A: This is exercise 1.8.25 from Probability and Random Processes by Grimmett and Stirzaker. The answer is 1/2 and can be shown to be true by induction.
