There are several applications of probability to attack deterministic problems. Generally, this approach is called the 'probabilistic method' or the probabilistic method of Erdos, who first applied it to solve a wide class of problems. For a survey of the technique, check out Noga Alon and Joel Spencer's book "The Probabilistic Method".

A specific example, due to Erdos, is an answer to Sidon's problem. Sidon asked whether or not one can find a set $B \subset \mathbb{N}$ such that $|B \cap [1, N]| = N^{1/2 + o(1)}$ for all $N$, and such that $2B = $ {$b_1 + b_2 : b_1, b_2 \in B$ is equal to $\mathbb{N}$. Erdos answered this question in the positive by using probabilistic arguments to show that such $B$'s exist. In particular, if one considers the random set $B$ defined by $\displaystyle p(x \in B) = \min\left(1, 10 \sqrt{\frac{\log x}{x}}\right)$, then $B$ satisfies Sidon's condition with probability 1. To date, no explicit example of such a $B$ is known to exist.