Yes, when creating a prize, you have to select a Recipient. It could be a rank, meaning the player who achieves that rank; or a bounty on a player, in which case you can choose "Random Player" and the TD software will choose a random player in your tournament on which to place the bounty (and the prize will be awarded to the hitman when the random player busts out); or a random player, meaning the TD software will simply choose someone in your tournament to receive the prize (and the prize will be awarded when the random player busts out); or it can be manual, in which case the TD software won't award the prize. In the last case you have to override the recipient to manually award the prize.
For the "Bounty on random player" and "Random player" recipient settings of a prize, the prize is awarded at bust-out. Therefore, you can create an event triggered by a random player being chosen for a prize. The event can be configured to display the randomly chosen player.