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The required probability = Probability of getting the desired configuration (8 black and 4 red) from a bowl having more black / probability of getting the desired configuration (8 black and 4 red) from ALL the bowls
Yes, I think the answer is good. By the way, for those of you who need to fine tune their probability/combination skills (and want that 700+ score), here is a link and a document giving very good explanations (especially the last 3 problems).