The problem is that pretty much all math on this site is either based around arguments over Shannon's number or some similar argument over the number of moves from a given position or number of positions.
I'd be much more interested to read about new attempts to disprove the Riemann Hypothesis or an elementary 4-color theorem but it's just not going to happen.
Impressive is a fascinating experiment from a couple of years back. This showed that trained chimps were better at certain types of short term memory games on a touch screen computer than humans. A video of this is well worth watching.
You left out a very important adjective. The experiment highlighted in the video demonstrateed that trained chimps were better at certain types of short term memory games on a touch screen computer than untrained humans.
Simplifying (2/32)^60 = 1/(2^64) or 1:2^64 or 1: (1.8 * 10^19).
I am curious if anyone else has another estimate, or finds a problem with or improvement to my calculation.
2/32^60 = 1/16 ^60. = .0625 ^60
http://en.wikipedia.org/wiki/Order_of_operations
I also find math more interesting than chimps. 
32^60 = 2,0370359763344860862684456884094e+90
1/ 2,0370359763344860862684456884094e+90 = 4,9090934652977265530957719549863e-91
4,9090934652977265530957719549863e-91 x 2 = 9,8181869305954531061915439099726e-91
1/16 = 0,0625
0,0625^60 = 5,6597994242666952296931995568049e-73
2/32^60 ≠ 1/16 ^60
Simplifying (2/32)^60 = 1/(2^64) or 1:2^64 or 1: (1.8 * 10^19).
I am curious if anyone else has another estimate, or finds a problem with or improvement to my calculation.
2/32^60 = 1/16 ^60. = .0625 ^60
what was wrong with the way gmillara wrote it? (2/32)^60 = (1/2)^64
That being said given the same condititions (an Infinite amount of time and an infinite amount of people) a human, who knows the rules of chess, would eventually defeat a computer that never makes mistakes.
...
That is only true if chess is forced win(which we don't know) from move 1, and if human is playing the side that has a forced win.
If not, you can not win if "computer never makes mistakes" (your postulate). 
My basic intuition tells me that the answer is absolutely zero chance and that any thought of there being the one time in infinity that blind luck defeats intelligence is nonsense. Is there a way to mathematically prove that the chances not even 1 in a gajillion to the billionth power? Perhaps we should be asking how many moves such a game would be likely to last. Suppose on average a game would last 12 moves. If it can be proven that 4 mistakes leads to definite defeat, and that a game will always last more than 4 moves does this prove that defeat is a mathematical certainty?
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.
As a theoretician, I think the approaches previously given are not quite correct. First, the probablity of some untrained winning against a GM is quite small and near enough to zero to be approximated that way. As the opponent rating drops at some point, the probablility approaches 0.5 where each player has equal chances to win in the long term. Based on chess rating systems, a 200 rating point difference is approximately quivalent to a multiplicative change in probably of winning of about 0.6. Assuming an untrained prlayer could master a rating of 400, then one would only need to consider the number of players in the world that are untrained in the world versus the number of players in the world that would have ratings above, say 1000. Against players above 1000, the probability of winning would to small to consider.
The world population is about 7 Billion. I would doubt tha the number of players in the world that could master a rating above 1000 is more than 100 million... therefore the probablility of an untrain player meeting a player with a rating greater than 1000 is about 1:70. Therefore, the probability of player winnign on chance only would be in the range of 0.45!
Of course locations will make a difference: if the player is in Russia, the probability would be much small than a player in one of the uncontacted tribes in the Amazon.
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.
...
almost certainly (p →1)before such thing would happen (chimp beats Magnus) some of the following things would happen:
1. chimp would die of old age
1. Magnus would die of old age
3. chimp would evolve
4. Earth would be destroyed by a Sun (http://en.wikipedia.org/wiki/Red_giant )
... + freely enter some other events in between
The chances of a primate (chimp, orangutan, gorilla, ...) defeating Carlsen are going to be higher if the primate has a +2000 ELO.
Anyway, more than saying if a chimp could defeat Carlsen, we'd better ask if a chimp could play an entire game against Carlsen. As said, a chimp is not going to learn all the legal moves in a week (not to talk about castling, promoting pawns and capturing 'en passant', between other things).
Humans are the last animals to have not solved chess. Monkeys find them amusing.
I would say the chances are 50-50 if both sides are playing by blind chance. However, playing blind chance (just making any legal move at random) against someone who knows what they're doing will probably give you about a billion to one shot of winning.
For example, in Chessmaster you can set up a computer personality to play basically random moves. You can play thousands of games against the computer and win every time. I'd say that it would be very easy for me to beat random chance moves a billion times in a row, and I'm not really that good at chess. Take someone rated 2000 or better, and they could go a lifetime without losing against random chance moves.
Playing an actual monkey, however, would be different because he could make you resign by flinging feces at you.
Chimpanzees are not the point. Why are you discussing chimpanzees?
Oh sorry. The original poster was talking about monkeys typing out the complete works of Shakespeare, not chimpanzees. My mistake. Perhaps monkeys would have a better chance, but then again their fingers are smaller than chimpanzees, so it might be harder for them to press down the keys especially if it was one of those old-style manual typewriters. Do monkeys even use typewriters any more? I thought most primates had switched to using wordprocessing programs for their desktop publishing needs. Or if all you want is a copy of the complete works of William Shakespeare, it might be cheaper to just buy them online. 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.
Nevertheless, I think it's easier for a chimp/monkey to solve the Rubik's cube than defeating Carlsen.
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.
Ravenoak has correctly spotted the error of those who claim that minor issues such as the chimps having no idea of the rules of chess, about pressing clocks and so on, are a problem. There is no need to resort to replacing traditional chess sets with a computer interface, as I suggested.
and again those who made a biggest error by forgetting that a failing to comply with a laws of chess would be a weighting factor in determining the outcome(chess vs. Shakespeare) would (still) rather prolong a discussion than accepting that they were wrong.
Dragec, it doesn't matter if 10^100-1 out of 10^100 chimps fail to comply with the laws of chess, one of the ones that do will play well enough to beat Carlsen. Think about it.
And if changing the postulate of the experiment, one might change a monkey for a human. Adding a computer interface with check for moves legality would alter the experiment.
As for the chimp vs. Magnus, probability is almost 1 that chimp would lose a game by failing to play a legal move, and pressing the clock.
Altering the experiment by adding various types of help to the chimp would lower that probability, but probably not visibly.
Again, having in mind players lifespan, probability is again close to 1 that players(with the added help) would die before a chimp would win a game. obviously the conclusion is also false as Magnus would be a monkey if he would let chimp play with him.
and those who are shown to be wrong, would rather argue indefinitely than accept they were wrong.