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The proportion of white balls converges almost surely to a random limit $\\Theta$, and $\\mathcal{D}=\\{\\Theta\\in\\{0,1\\}\\}$ denotes the event when one of the colours dominates. The phase transition, in terms of the sequence $(\\sigma_n)$, between the regimes ${\\mathbb P}(\\mathcal{D})=1$ and ${\\mathbb P}(\\mathcal{D})<1$ was obtained by R. Pemantle in 1990. 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