You click the dead thread link. Roll dice. Bad roll, no egg.
You click the active thread link. Roll dice. Bad roll, no egg.
You click hospital. Roll dice. Bad roll, no egg.
You click jail. Roll dice. Bad roll, no egg.
You click city. Roll dice. Bad roll, no egg.
You click the dead thread link. Roll dice. Good roll, yay egg.
Edit: There is an anti-cheat system that affects the % chance of an egg appearing. We know this. This would imply that sitting reloading the page or something else that could be consider kinda dodgy might effect the chances of an egg spawning on page load, but we simply do not know for certain.