Interesting fact: you can choose a random element from a sequence of unknown size in a single pass. The cost is having to always iterate over the entire sequence, and generating O(n) random numbers instead of O(1).

The algorithm is quite simple:

    var choice = elements.First()
    var n = 1
    for e in elements.Skip(1):
       n += 1
       if rand(n) == 0:
          choice = e

The algorithm is correct because, at each stage, each of the preceding elements is chosen with probability 1/n. The invariant is maintained because the next element is chosen with probability 1/(n+1) and therefore scales the preceding probabilities by n/(n+1) giving 1/(n+1).