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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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    Repeating a tag in the title is redundant since the tags appear wherever the title appears. The tags indicate the general field of the question, and the title should more specifically summarize the question, e.g. "discarding balls with repeated colours while drawing from an urn"2012-09-24
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    I think the drawing procedure of the balls is irrelevant. All that matters is how many of each type you have at the beginning. Thus the probability of the last ball being azure will be $n_a/n$ or $3/8$ for a starting configuration like in your example.2012-09-24
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    @Raskolnikov: Not true IMHO, try simulate it with poker cards, you'll find that even if you begin with 10 azure and 1 carmine, it's not that easy for the last one to be azure.2012-09-24
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    That's what I did, I simulated it. But apparantly, 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

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