There are lots of examples of this phenomenon connected with "the probabilistic method". There the problem is to prove a certain type of object exists (there is at least one piece of hay in the stack). But it's hard to build one. So what you do is construct one randomly, then prove that on average or with high probability, your construction satisfies the requirements. This proves that at least one thing in the stack is hay, or actually that on average or most of the stack is hay.

I'm thinking of graphs, for example expander graphs; error-correcting codes; and probably lots more I'm forgetting. In these cases it then becomes a research program to construct such objects explicitly (which has a lot of history with expanders and codes).