I have recently played the game 2048, created by Gabriele Cirulli, which is fun. I suggest trying if you have not. But my brother posed this question to me about the game:
If he were to write a script that made random moves in the game 2048, what is the probability that it would win the game?
Combinatorics is not my area, so I did not even attempt to answer this, knowing this seems like a difficult question to answer. But I thought someone here might have a good idea.
Also, since we are not concerned with time, just with winning, we can assume that every random move actually results in a tile moving.
While the answers below shed light on the problem, only BoZenKhaa came close to providing a probability, even if it was an upper bound. So I would like to modify the question to:
Can we find decent upper and lower bounds for this probability?
I implemented a simulation of 2048, because I wanted to analyze different strategies.
Unsurprisingly, the result is that moving at random is a really bad strategy.
Above you can see the scores of 1000000 random games (edit: updated after bugfix, thanks to misof). The score is defined as the sum of all numbers generated by merge. It can be viewed as a measure of how far you make it in the game. For a win you need a score of at least 16384. You can see that most games end in a region below 2000, that is they generate at most a 128-tile and loose subsequently. The heap on the right at 2500 represents those games that manage to generate a 256 tile – those games are rather rare. No game made it to the 1024-tile.
Upon request, here is the plot for highest number on a tile:
When it comes to “dumb strategies”, you get better results cycling moves deterministically: move up, right, up, left and repeat. This improves the expected highest number by one tile.