What is my chance of beating a Grandmaster

If the GM is of sound mind  and you give him/her  at least 15 minutes for the game, the probability of you defeating the GM is about 1 in a million.     

However, if the GM has less than 2 minutes for the entire game, then you stand a chance of them screwing up.   That is about the only type chance you have,ie; you will not outplay them.    they have to make a major blunder.

When World Champion to be Capablanca was 4 years and 10 months old, he was given Queen odds by a Master player.    Jose won without a problem.    

http://www.chessgames.com/perl/chessgame?gid=1481959

Handicap and odds are two different things.

First of all, the value of a handicap depends on the strength of the player receiving the handicap. For example playing a beginner down a queen may only boost their rating 50 points, but playing a GM down a pawn is easily worth a few 100 rating points.

A knight to an 1800 player is about 400-500 points, so you could win a game, but you'd probably lose a match.

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Odds are actually built into the rating system... although they're not accurate in the extremes, you'd have less than 1% chance of winning a match.

I'm not a statistician, and haven't studied mathematics since I went to school, but I dare say the chances of an E1800 player defeating an E2500 grandmaster are significantly higher than about 1 in a million - at normal time controls. When I was about E1800, I defeated an IM rated about E2400, and in that same tournament there was a GM who lost against an E1900. (Now, a GM with anything between 2 to 15 minutes on the clock, against an opponent playing say 40/2h, is likely to lose a great many games.)

The wide drawing margin in chess means that in order to win a game your opponent will have to make quite a number of minor slips or drop a real clanger. However, this obviously works both ways. A GM is not interested in drawing against an 1800 player, and therefore is likely to try very hard to win, sometimes leading to him/her overpressing.

Yeah, the chances are better than one in a million.

For example

http://www.chessgames.com/perl/chessgame?gid=1377482

Someone (Jeff Sonas?) a few years ago compared the expectancy against actual results, and found that the expectancy was off by quite some margin. The lower rated player tended to do better than expected, and the deviation was even more marked at more extreme rating differences. This was published on ChessBase, but I can't find the article right now.

Don't remember, but I think so. I know that they've changed the k factor in recent years.

In 2001, Terry Chapman, E2200-2250 at the time, played a four-game charity match Gary Kasparov with the former world champ giving him the odds of two pawns. Kasparov only won 2½-1½ (+2, -1, =1). http://www.chessgames.com/perl/chess.pl?pid=58505&pid2=15940

Chapman was an England junior squad member in the 1970s (and nowadays he's a regular member of the England teams competing in the World and European Senior Teams Championships), so clearly not a mug by any means, but Kasparov was still rated some 600 points higher. It doesn't actually take much in terms of material to even out the odds considerably.

Your odd of winning against +665 elo higher rated opponent is 1 in 46. ( excluding the draws). The odds can be higher up to 1:100 or 1:200 if we include the draws.

Your chance of pairing against a GM in a tournment will be in 9 to 11 round swiss system. So everyone is taking risks( risky opening, complicated lines) to beat everyone to get the prizes.

Realistically your odd can be 1:100 .( I need to get  draw odd data to calculate accurately).

As I explained above, that odd exclude draw.

In fact if we include draw ratio, ( lets us says draw odds among 2800 GM is 90%), then a 2800 rated GM beating another 2800 become 5% chance, rather than 50%.

Apart from the fact that we're talking the world champion, and one of the world's best blitz players, arguably rook odds are better for the player giving the odds than the one receiving it. Rooks tend to enter the game much longer after bishops and knights, and typically don't come into their own until the endgame. I strongly suspect that Trent would have had a much better chance had Carlsen instead given odds of just a bishop or a knight. Had Trent been given the odds of bishop and knight (by 'point count' only marginally better than a rook, but vastly superior in the middle-game) he probably would have won 5-0.

There are mathematical formulas that can be more precise than all the speculation and who-ha being offered up by well wishing but Naive players.

This is not exact, but close enough. a 200 point rating difference gives the higher rated player about a 75% chance of winning any particular game. Therefore the probability of a higher rated player winning if they are rated 800 points higher is approximately 1 - (1/4)^4 or .996. 

Meaning the lower rated player has a win expectancy of about .004. This means you would be expected to win one game (or get two draws) for every 250 games played. 

so your chances of a win are actually a bit higher than some of the folks allege. 

Morphys-Revenge: Are you sure that's the current expectancy? That sounds very much like the original calculations by prof. Elo, and they've been revised numerous times since they were introduced.