Elroch, I hope you find the thread as amusing as I do, we're just wasting time.
Don't take it seriously, anything can happen if you put the infinity in equation.
Hey, chimps even became humans(according to some) and it took only a little time (of Earth's life) 
Our human intuition almost invariably fails us where the concept of infinity is concerned.
Not only is defeat over infinity trials not a mathematical certainty...the opposite is quite unassailably true. Given enough trials, not only is victory assured, but literally infinite victories are assured.
You don't need to limit the interface to a chimp-friendly software screen. You don't even need to give the chimps the most rudimentary possible training in piece manipulation.
Just stick a chimp in a room with Magnus, infinity times.
The chimp's random actions will eventually produce legal chess moves. More rarely, they will eventually produce a string of consecutive legal chess moves. More rarely still, they will eventually produce a string of consecutive perfect chess moves.
Randomness, spread over infinite trials, brings all possible results.
What about the main question: what is easier for monkeys? to defeat Carlsen or to write a Shakespeare?
For this question it is very important to know if the monkey would do random legal moves, random any moves or even totaly random activity, as urinating in the board.
I believe that we should be rational in the question. To assume the monkey would do legal moves is the same to assume they know English vocabulary and grammar, making both activities much easier.
If we assume totaly random activity in any of the cases I am sure that chess is more difficult. Typing is just pressing stuff while to play chess the monkey has actually to be patient and move only one piece per time, waiting for the oponent's move. That is way more unlikely than just pressing random stuff.
There's a good chance that Carlsen would be dead by the time the chimps had made a recordable move!
For an infinity trial we must assume an immortal Carlsen, in the same way the monkeys should be immortal too.
So then, Fezzik, we should do the same with a touch-screen where the monkey is rewarded every time he writes some coherent word. 
... probability is almost 1 ...
Yes, just like the probability is almost 1 that a chimp will fail to write the complete works of Shakespeare.
http://en.wikipedia.org/wiki/Infinite_monkey_theorem
"In 2003, lecturers and students from the University of Plymouth MediaLab Arts course used a £2,000 grant from the Arts Council to study the literary output of real monkeys. They left a computer keyboard in the enclosure of six Celebes Crested Macaques in Paignton Zoo in Devon in England for a month, with a radio link to broadcast the results on a website. One researcher, Mike Phillips, defended the expenditure as being cheaper than reality TV and still "very stimulating and fascinating viewing".
Not only did the monkeys produce nothing but five pages consisting largely of the letter S, the lead male began by bashing the keyboard with a stone, and the monkeys continued by urinating and defecating on it. Phillips said that the artist-funded project was primarily performance art, and they had learned "an awful lot" from it. He concluded that monkeys are not random generators. They're more complex than that. … They were quite interested in the screen, and they saw that when they typed a letter, something happened. There was a level of intention there."
... probability is almost 1 ...
Yes, just like the probability is almost 1 that a chimp will fail to write the complete works of Shakespeare.
Yes, this is (again) extract from wiki(just for Hamlet), showing the "futility" of the discussion (and again, monkeys are not a random generators) :
Ignoring punctuation, spacing, and capitalization, a monkey typing letters uniformly at random has a chance of one in 26 of correctly typing the first letter of Hamlet. It has a chance of one in 676 (26 × 26) of typing the first two letters. Because the probability shrinks exponentially, at 20 letters it already has only a chance of one in 2620 = 19,928,148,895,209,409,152,340,197,376 (almost 2 × 1028). In the case of the entire text of Hamlet, the probabilities are so vanishingly small they can barely be conceived in human terms. The text of Hamlet contains approximately 130,000 letters.[note 3] Thus there is a probability of one in 3.4 × 10183,946 to get the text right at the first trial. The average number of letters that needs to be typed until the text appears is also 3.4 × 10183,946,[note 4] or including punctuation, 4.4 × 10360,783.[note 5]
Even if the observable universe were filled with monkeys typing from now until the heat death of the universe, their total probability to produce a single instance of Hamlet would still be less than one in 10183,800. As Kittel and Kroemer put it, "The probability of Hamlet is therefore zero in any operational sense of an event…", and the statement that the monkeys must eventually succeed "gives a misleading conclusion about very, very large numbers." This is from their textbook on thermodynamics, the field whose statistical foundations motivated the first known expositions of typing monkeys."
Even if the observable universe were filled with monkeys typing from now until the heat death of the universe, their total probability to produce a single instance of Hamlet would still be less than one in 10^183,800. As Kittel and Kroemer put it, "The probability of Hamlet is therefore zero in any operational sense of an event…", and the statement that the monkeys must eventually succeed "gives a misleading conclusion about very, very large numbers." This is from their textbook on thermodynamics, the field whose statistical foundations motivated the first known expositions of typing monkeys."
There continue to be two arguments going on simultaneously.
In one, we have "in any operational sense," as discussed by the researchers. Where they're saying that given observed behavior, even billions upon billions upon billions of monkeys, performing for billions upon billions upon billions of years, would still have a chance of writing Hamlet OR defeating Carlsen that would be so slim as to be essentially zero. That the odds are literally so slight, that given the projected remaining life span of the universe as we know it, we might as well call it zero.
Then we have the scenario with infinite trials.
Even the unspeakably mind-boggling numbers of attempts the researchers are talking about in "observational senses" round to zero when compared to infinity.
It did not take so long for a human to evolve, so if infinity is involved, chances are indeed higher that monkeys will sooner evolve and play the chess correctly, rather than win just by a random moves(and Monkeys are not a random generators):
http://en.wikipedia.org/wiki/Timeline_of_human_evolution
Even the unspeakably mind-boggling numbers of attempts the researchers are talking about in "observational senses" round to zero when compared to infinity.
I was with you until that last statement. I don't know what it means to round a number to 0 when comparing it to infinity.
could be derived from this.
All monkeys are primates.
Magnus Carlsen is a primate.
Therefore, Magnus Carlsen is a monkey.
You could probably get a Kindle edition for a dollar or two, and save the whole hassle of checking over the monkeys' handiwork for typos and spelling errors.
I have the complete works of bill on my ipad for free.
If the original poster wants a copy of the complete works of Shakespeare, perhaps it would be simplest just to ask Ozzie if he can borrow his ipad for a while.
Ha ha you guys are very funny.
I suppose I should have stated in the original question that there is some mechanism by which the monkey is able to make only legal moves so he can't lose by some silly technicality, just as the monkey in front of the typewriter can only press one of a group of keys (though I guess he can still type gobbledegook).
Perhaps the ape is sitting in front of a touchscreen where the current set of legal moves is displayed randomly on the screen and the monkey just pokes at one. I doubt that a monkey sitting in front a real chess board would be able to resist sucking on a knight or trying to push a bishop up his bum (I've seen what the monkeys get up to at the zoo).
The chimpanzee/Shakespeare experiment thing is covered in a hilarious short story "Inflexible Logic" by Russell Maloney, where 6 chimpanzees are put in a room with 6 typewriters and an unlimited supply of paper and bananas. I won't spoil the story by telling you what happens but it is not what you expect! It's included in a book called Fantasia Mathematica (along with lots of other funny science- and maths-based stories).
arkledale
http://en.wikipedia.org/wiki/Infinite_monkey_theorem
"In 2003, lecturers and students from the University of Plymouth MediaLab Arts course used a £2,000 grant from the Arts Council to study the literary output of real monkeys. They left a computer keyboard in the enclosure of six Celebes Crested Macaques in Paignton Zoo in Devon in England for a month, with a radio link to broadcast the results on a website. One researcher, Mike Phillips, defended the expenditure as being cheaper than reality TV and still "very stimulating and fascinating viewing".
Not only did the monkeys produce nothing but five pages consisting largely of the letter S, the lead male began by bashing the keyboard with a stone, and the monkeys continued by urinating and defecating on it. Phillips said that the artist-funded project was primarily performance art, and they had learned "an awful lot" from it. He concluded that monkeys are not random generators. They're more complex than that. … They were quite interested in the screen, and they saw that when they typed a letter, something happened. There was a level of intention there."
With regard to the quote from Mike Phillips above, I don't see any way to distinguish this experiment from reality TV.
Yes. Next question?
There are numerous factors to take into consideration. Assume the chimp is playing on a computer screen that does not allow illegal moves to be played, and the clock is not a factor.
Just making it through the first 5 moves without falling behind badly could take thousands of random games.
If the chimp is allowed to throw his poop at magnus the odds are in his favor.
If you're dealing with infinite time, you only need one monkey, not infinite monkeys.
While I haven't done the math, don't care to do the math, and really have no interest in this other than to note that it is rather insulting to monkees, the number of monkees can matter (not saying it does in this case or not). There are different infinities. The number of integers, for example, is one level of inifity, but the number of ordinal numbers is a completely different, bigger level of infinity.
http://en.wikipedia.org/wiki/Aleph_number
Duncan Suttles has won many a game with random moves.