One nice example is Bernstein's proof of the Weierstrass theorem. This proof analyses a simple game: Let $f$ be a continuous function on $[0,1]$, and run $n$ independent yes/no experiments in which the “yes” probability is $x$. Pay the gambler $f(m/n)$ if the answer “yes” comes up $m$ times. The gambler's expected gain from this is, of course, $$p_n(x)=\sum_{k=0}^n f(k/n)\binom{n}{k}x^k(1-x)^{1-k}$$ (known as the Bernstein polynomial). The analysis shows that $p_n(x)\to f(x)$ uniformly.
S. N. Bernstein, A demonstration of the Weierstrass theorem based on the theory of probability, first published (in French) in 1912. It has been reprinted in Math. Scientist 29 (2004) 127–128 (MR2102260).