How do I fetch a post ID randomly from published posts?

I am currently using the Get-a-post plugin which allows me to put a single post in the header based on the post number I choose. This is what I have:

<?php get_a_post(234); ?> which pulls the post with the_id() of 234. I then added the following code:

<?php srand(time());
$random = (rand()%319); ?>

and replaced <?php get_a_post(234); ?> with <?php get_a_post($random); ?>

This is all fine and dandy, but when the random number generator picks a number from a post that is not published and may or may not exist, the results are blank.

Is there a way to define the array from which the random numbers can be chosen that is drawn directly from the database such that if a post is added or removed the script will adjust?