My app Rock, Paper, Scissors, Lizard, Spook has an AI that uses pattern recognition and basic probability. If the human player has a simple pattern, the AI will recognize it and play accordingly.
My app The Oracle uses randomness to keep the user more entertained. Not really AI, but it shows how a little randomness in a very simple toy can be entertaining.
I've written a Battleship game that uses more sophisticated AI with rule based AI and probability maps. If the computer hits a ship, it knows to continue shooting around that hit. If the computer is looking for a ship, it will calculate the location that will most likely contain a ship. The AI is good enough to beat a human playing the best they can if the chips fall in the right way - just as human vs. human players win. Now I just have to write an app for it!