Devise a combinatorial problem in which it is easier to solve via probability theory than by counting methods We easily find probabilities by counting the number of ways we can meet a condition and dividing it by the number of total possible outcomes.  This is using combinatorics to solve probabilities.
Im looking for a scenario, or a type of complicated combinatorial problem, in which the approach at counting the number of ways of meeting a condition is actually easier by multiplying the probability of meeting the condition (determined in other ways) by the total possible number of outcomes.
I realize this may be an absurd question, but its been in the back of my mind for some time.  Can anyone imagine up such a problem?  This is for the sake of education, so that we may all learn of new problem solving techniques.
 A: Probability theory has been applied to combinatorics for a long time. There is a rich literature for you to explore.
Here is a brief expository paper. I recommend skipping section 1 and going immediately to section 2, then returning to 1 once the method is clear to you.
A commenter (Quinn Culver) mentioned the Wikipedia article on the probabilistic method, which seems unusually good. I generally don't like to just spam Wikipedia entries as answers, but it contains some nice examples.
For more, see the book published on this topic.
Here is an example problem, from the expository paper:
Let $A$ be any set of $n$ residues mod $n^2$. Show that there is a set $B$ of $n$ residues mod $n^2$ such that at least half of the residues mod $n^2$ can be written as $a + b$ with $a \in A$ and $b\in B$.
A: There is a very nice proof of Hook Length Formula. Its basic idea is to create an algorithm that generates random instances with uniform (discrete) distribution  and use $\frac{1}{p}$ for counting. Take a look at 

C. Greene, A. Nijenhuis, H. S. Wilf, “A probabilistic proof of a formula for
  the number of Young tableaux of a given shape”, Adv. in Math. 31
  (1979), 104-109,

and

Jean-Christophe Novelli, Igor Pak, Alexander V. Stoyanovskii, “A direct
  bijective proof of the hook-length formula”.

I hope it is an example of what you asking for ;-)
A: You can look at this problem:
Combinatorial proof involving factorials
And my solution to this problem using Probability.
https://math.stackexchange.com/a/298534/48639
