As Tsuyoshi Ito commented, the probabilistic method in combinatorics is an example. You use a probability measure on the space of possibilities, and show that the set with the desired probabilities has positive measure, hence is nonempty.

The classic example of this is the result by Erdős (1947) that the Ramsey number $R(t,t)$ grows at least exponentially with $t$. If you consider a random coloring of the edges of the complete graph on $n$ vertices, then the probability that a particular complete subgraph on $t$ vertices is monochromatic is $2^{1-{t \choose 2}}$. If ${n\choose t} 2^{1-{t \choose 2}} \lt 1$, then a random coloring has no monochromatic subgraph with positive probability. This is the case for $n = \sqrt 2^t$, $t\gt 2$, so $R(t,t) \ge \sqrt2^t$ for $t \gt 2$.

For slightly smaller $n$, most random colorings of the complete graph on $n$ vertices have no monochromatic subgraph of size $t$, but finding a construction has been an open problem.