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In enumerative combinatorics, the bijective approach has proved extremely successful in studying diverse combinatorial objects, such as trees, partitions, Young tableaux, etc. However, in the last two decades, a number of probabilistic algorithms for random generation revealed a new and often unexpected side of these classical objects. In the first half of the talk, we present a brief survey of the probabilistic random generation, together with some applications. In the second half of the talk, we introduce a new weighted hook walk giving an interesting (non-uniform) distribution on standard Young tableaux. We then show how to go back and obtain a new short bijective proof of the hook-length formula. This is a joint work with Ionut Ciocan-Fontanine and Matjaz Konvalinka. The talk will be accessible to a general audience, so no background in combinatorics is assumed.