How do they decide which bronc a cowboy rides?

Because this is the kind of thing that would come up in a conversation - bucking horses are owned or leased by contractors, so how do they decide which cowboy rides which horse?

The answer is a random draw (which is also done to assign calfs and steers to ropers and wrestlers). Draws are often done using computers these days and posted to the internet before the rodeo. Obviously, lower tech methods are sometimes still used, but the idea is that the "draw" should be random.