If you're a developer looking to get some exposure for your high-quality app, or if you're a user hunting for new apps and would like a chance to win a paid app at no cost, then Jack Underwood's new Promo Codes app may get you there, which you can install for free from the Google Play Store.
UPDATE: This app is no longer available in the Google Play Store.
Promo Codes will give you a chance to win an app every 12 hours (from your last attempt). Underwood says that the odds of winning are "low, but fair" and that there's very little chance that you'll be a winner in the first few days that you use the app.
Simply open the app to see the featured paid app, click the "I'm Feeling Lucky" button, then see if you've won.
Promo Codes users should get used to seeing the "Unlucky!" screen. Maybe your luck will change in 12 hours, though.
From the sound of the description on the Play Store page, Promo Codes is more concerned with giving users a look at apps from up-and-coming developers, and not giving them away for free. As Underwood bluntly states: "If you're expecting to win often, you're wasting your time downloading Promo Codes."
On the plus side, you don't need to make a username or sign up for anything, so there's really nothing at stake. And if you don't win, you could come across an app that you think is worth spending a couple bucks on. Promo Codes sends your phone a notification when your 12 hours are up, so you can check out another app that you will (most likely) not win.
If you're a developer and would like to have your app promoted inside Promo Codes, email email@example.com with your request.