TITLE: Probability that random moves in the game 2048 will win
QUESTION [62 upvotes]: 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. 
Addendum
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?

REPLY [4 votes]: $3.3\times10^{-32}$
This result has been achieved using 5 runs of branching simulation in the following configuration: Run 16777216 simultaneous game simulations. Whenever less than half of them remain make a clone of every remaining simulation. Weigh the result of every simulation relative to what "generation" it end in, so simulations ending before the first branch will count for $1/16777216$. After one branch it is $1/33554432$, after two $1/67108864$ and so forth.
This method ensures that every original simulation has the same weight in the final result, while the simulation keeps running long past the point where standard simulations would have stopped.
It should be obvious that given enough runs of such a simulation, the average result will converge towards the answer. The less obvious part is that it does so much faster than a standard simulation. This is only possible due to nature of the game and the question asked, the following are crucial properties of the question:

We are searching for a very low probability.
The variation in probability for each node of the simulation to succeed is limited, i.e. the game has no "super states" where the probability of success is abnormally large relative to the average game state at the same level of progress.
There is an easily identifiable failure state that can be used for pruning the collection of simulations.

With these constraints in mind, the game and question seem perfect for the solution method. While there is significant variation in probability of success between nodes, it is not so great that a few nodes can dominate in a collection of several million.
So far the only means for estimating the precision of the result that I have is comparing the results of different runs. The results of 5 runs:
$$3.29\times10^{-32}$$
$$3.43\times10^{-32}$$
$$3.33\times10^{-32}$$
$$3.30\times10^{-32}$$
$$3.31\times10^{-32}$$
I'll try to get some source code online tomorrow.