[pzhon]

That is a nice inequality.

Here are a few applications of probability to other parts of mathematics:

[img]/images/graemlins/diamond.gif[/img] The Stone-Weierstrass Approximation Theorem
Every continuous funtion f(x) on the interval [0,1] can be uniformly approximated by polynomials f_1(x), f_2(x), f_3(x), etc. That is, for any small number z>0, you can find some n so that for all x, -z < f(x)-f_n(x) < z. This is an important fundamental result in analysis.

You can prove this by constructing the Bernstein polynomials:
f_n(x) = Expected value of f(X(x,n)/n),
where X(x,n) is a sum of n independent 0-1 random variables which are 1 with probability x.

[img]/images/graemlins/diamond.gif[/img] The probabilistic method in combinatorics.
This is a nonconstructive technique for showing that combinatorial objects with certain properties exist. The idea is to construct probability measures on collections of objects, then use simple results or calculations in probability to show that the probability of encountering a structure with the desired property is greater than 0. This technique is due to Paul Erdo"s.

Here is a probabilistic proof that there are infinitely many primes: For any finite list of primes p1...pn, find the probability that a random integer on a large interval is not divisible by any of these primes: (1-1/p1)(1-1/p2)...(1-1/pn). This is greater than 0, so there is some integer in that interval not divisible by any of these primes.

Packing unit spheres densely in higher dimensions is important for coding theory (ideal transmission of information, not cryptography). In 2 dimensions, we know the best packing for unit circles: Every circle has 6 neighbors. In 3 dimensions, there is more flexibility, but it has been shown recently (Hales) that the most natural "cannon ball" packing has the highest density. In dimensions up to 24, we know some fairly good packings of spheres. In many more dimensions, say 10,000, we don't have any reasonable candidates for the densest packing of spheres. The unit 10,000-dimensional cube has room for a lot of unit diameter spheres (with wrap-around), and explicit constructions are poor. In fact, we can prove that denser packings exist by throwing in a lot of centers of spheres randomly, then estimating the probability that at least two are within didstance 1 or each other. If the probability is less than 1, there is some way of packing that many spheres without overlap.

[img]/images/graemlins/diamond.gif[/img] Random Matrices.
Much of quantum mechanics in practice involves linear algebra. If you study a hydrogen atom, this is not too hard. If you study a uranium atom, this is extremely complicated. Even for a simple model of the lower energy states, you may have to understand the eigenvalues of matrix with hundreds of dimensions, whose coefficients you can't estimate very easily with accuracy. People decided to analyze the properties of random matrices, to tell what can be expected of a typical matrix, a typical symmetric matrix, or a typical Hermitian matrix. Then you hope the specific matrix corresponding to uranium isn't particularly unusual.

Actually, I haven't seen published that many spectral properties of random matrices are really just properties of random polynomials. After all, the characteristic polynomial of a random matrix is a random polynomial.