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Suppose we have a barrel with three different kinds of marbles: red, green and blue. The probability to find a red marble in the barrel is R0, analogously the probability for green is G0 and for blue is B0. We pour some of the marbles in a smaller bucket. Now the probabilities in the bucket are R1, G1 and B1. Finally, we pour from the bucket some marbles in a glass. The probabilities in the glass are R2, G2 and B2.

What are the expected values for R2, G2 and B2?

I can solve this problem if I know R0, G0, B0, G1 and B2, and consider the information about Xi (where X can be R, G or B) that I obtain from Xi+1 negligible.But what can I say if this information isn't negligible? Or if I only know R0, G1, B2?

Thanks a lot. I don't even know what can I read to solve the problem.

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Apparently the first step would be to state the problem in a meaningful way. Right now it hardly makes much sense from a formal standpoint (meaning that you can interpret it in many different ways and get different answers for different interpretations). –  fedja Jun 17 '11 at 16:57
    
I would assume 1) "To find a red marble" means that you are selecting a marble at random from the population, each marble having the same probability of being chosen. Thus these probabilities Ri, Gi, Bi are simply the fractions of marbles in the barrel, bucket or glass that have the given colour. 2) In the pourings each marble has the same probability of ending up in the bucket or glass. Thus knowing R0, G0, B0 (but not R1, G1, B1), the expected values of R2, G2, B2 would still be R0, G0, B0. If you don't know R0, G0, B0, in order to proceed in a Bayesian fashion you need to assume a prior –  Robert Israel Jun 17 '11 at 17:54
    
(continued from previous comment) distribution for these, as well as something about the total numbers of marbles in the barrel, bucket and glass. –  Robert Israel Jun 17 '11 at 17:57
    
Fedja, I'm sorry for being not enough precise, I'm not a mathematician. Robert Israel better described it, thanks. Excuse me my ignorance Robert, but if I want to know the final expected values, do I need a prior or should I consider all the possible combinations? (Sorry again for my informality) –  Natxo Jun 17 '11 at 18:50
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