It is not clear (to me) if you want to start fading after half a second, or fade in in half a second. However, going with fade in after half a second. If you want to do it otherwise, just ignore the stuff with setTimeout()
The general overview of what you want to do is:
Create a function when the page has loaded that is called after half a second (setTimeout)
That function should generate a random number, with the min as 0, and the max as the number of children of the #container element minus 1
Fade the child of the #container with the index supplied by the random number.
Pusdo code (It is a long time since I have done anything with jQuery. Or Javascript for that matter)
function onDocumentReady(){
setTimeout(500, "fadeInRandom()");
}
function fadeInRandom(){
var numElements = $("#container").children().length;
var randomElem = Math.random() * (numElements-1);
$("#container").children()[randomElem].fadeIn();
}