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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.

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    Th$a$t's what I did, I simulated it. But appara$n$tly, I have made a mistake in my first attempt because I'm trying again now and obtaining a different result. So forget my earlier comment.2012-09-24

2 Answers 2

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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.

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    Chris' interpretation is correct. If there's$a$ball in the urn, you have to keep drawing.2012-09-24
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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.