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Assume that ${\\bf A}:=(A_n)_{n\\in{\\Bbb N}}$ is a sequence of Lebesgue measurable subsets of ${\\pmb M}$ satisfying a necessary density condition and ${\\bf x}:=(x_n)_{n\\in {\\Bbb N}}$ is a sequence of independent random variables which are distributed on ${\\pmb K}$ according to a measure which is not purely singular with respect to the Riemann volume. We give a formula for the almost sure value of the Hausdorff dimension of random covering sets ${\\bf E}({\\bf x},{\\bf A}):=\\limsup_{n\\to\\infty}A_n(x_n)\\subset {\\pmb N}$. 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