Very poor, around 10:1 (or 1:10, whichever one is bad). I know the comparison is overused, but I think it is especially apt in this context: Imagine asking someone if the US could lose a war with Vietnam in the 60s. It was mostly external constraints in both contexts that prevented the superpower from dominating. The British did not want to alienate the colonists, as they were one of their most valuable trading partners and retaining the colonies after raping and pillaging would have served little purpose. So they held back, appointed colonial sympathizers to positions of power in the operation, and in general did not try to win at all costs. The involvement of the French, both in Europe and North America, tipped the scales considerably in the direction of the Americans. If someone asked in 1770 who would win between the American colonies and the British Empire, most people would have taken pretty much any odds you offered on Britain.

I'm going to go against the grain here. I think the odds would have been fairly even and a slight edge could have gone either way. Why? Well this has nothing to do with history at all but instead has to do with how a bookmaker sets odds.

Let's say I were to set the original odds at 1:10 like another commenter has suggested. The ramification of that is heavy betting on the favorite... and betting trends will then further drive the odds one way or the other to protect the book. For example, if they were 1:10 and everyone was betting on America anyway then the odds might change to 1:10, making it more less enticing for someone to bet on America.

So now we get to the initial odds. America is vastly underpowered, but this really doesn't matter to me much (the book). I see that England is across the ocean and we live in an age before flight. I also see that America is territorially vast. In addition to this, I see that France is seriously pissed off at England after being defeated in the Seven Years' War.

So let's say we're having this discussion in 1764.

We don't know if France is going to get involved or not, but we do know that just strictly based on geography that it is essentially impossible for England to hold the colonies as an imperial possession indefinitely. It was just never going to happen. Maybe they could have won the revolutionary war and forced a peace and gotten some favorable terms for a few decades, but as the American population grew and the territory itself grew the idea that it was going to remain in English control just becomes silly. At some point, regardless of whether America won independence or not, at some point America would start to, "possess sovereignty" which is an esoteric and abstract sort of concept. One may argue that America possessed sovereignty in 1764 and that it was largely understood that a military conflict with England was inevitable. You can probably find dissenting opinions here if you look at the writings of the time, but you also have groups like the Sons of Liberty popping up in 1765.

Also of importance here is the defeat of France in the French and Indian War and the Seven Year's War.... which are basically the same war being fought on two different fronts. The point here is two-fold: 1) The colonies now had combat experience, and armaments. In fact, this is where George Washington's military career began. 2) France was seriously pissed off at England.

In retrospect if you take everything into consideration, especially the French assistance (which was not guaranteed in 1764) then I think the odds would have either been close to even, or slightly in America's favor and for my justification here I'll defer to the idea that it was really just impossible to think of England maintaining sovereignty over the growing population of the colonies that were an ocean away. The colonies weren't like a little island or something, and the American territory is vast. The idea (in 1764) that the Americans were going to fight a standard war and not make use of cover, etc., is really kind of a laughable and silly idea.

So really the entire question to me comes down to the likelihood that France was going to come America's assistance. In 1764 if I believed it to be a very low probability, then I would have set the original odds in England's favor, but not heavily. This has nothing to do with the odds that England would win, but everything to do with the way that odds are set to protect the house.

On the other hand if I could reasonably expect France to get involved then I would have set the original odds in America's favor.

Geopolitics are really the most important consideration here. Another commenter said, "Imagine asking someone if the US could lose a war with Vietnam in the 60s" with respects to what the odds would have been. Well, it would have depended on who you asked. If you asked General Giap he might have told you that if you killed 10 of his men for every 1 of yours then Vietnam was going to win. Now lets look at the numbers. 58,000 Americans died and (according to Vietnam) there were 1.1 million deaths. So Giap was actually being quite conservative with his claim because in reality we killed more than 20 of his men for every 1 he killed of ours and yet they still won.

Now, let's re-imagine Vietnam without any Chinese or Russian support to the Vietnamese. Or you could re-imagine it in the context that the US invaded North Vietnam and tried to occupy it. These are the real factors that won or lost the war.

Flipping to the American revolution there isn't a lot that the English could have done different (I don't think, feel free to ask someone who has studied British military tactics of the time.) The only real factor here is France. So as a bookie you'd have to evaluate all of these things and set the odds yourself. In 1764 if you reasonably believed France was going to get involved then I think that America would have had the edge... this would likely have resulted in big bets on England... which would likely have resulted in big profits for you.

You're a bookmaker. You aren't trying to guess who is going to win. You're trying to make money no matter what. Even if you guess wrong, and then as people bet you change the odds to reflect it. For example if they start out 1:1 and everyone is betting on England, then they become 1:2, and so forth. Eventually the odds become so great that people will start to bet on the underdog just because of how much they can win

In my own estimation? I would have bet on France getting involved and therefore would have given the edge to America simply because of the sheer impracticality of maintaining indefinite power over the colonies.

EDIT: Imagine it's a game of Civilization. You're across the water and have great defensive positions/units. There is no real technology gap unless you want to talk about naval warfare... which is really irrelevant to the larger war and need for ground occupation (see the war in Vietnam). In fact, the Americans maybe have a slight edge on the ground because their units can move through forests at twice the speed of your units. You're pretty sure you're going to get crushed when all of a sudden the huge empire next to England declares war and gets involved. I mean, if you know how to play Civ... you win that encounter every time. So now the discussion has moved from geopolitics to the competency of leadership and again it has to be remembered that by 1764 that the colonies had "some" military experience.

People are way to generous on here. The odds on my opinion were something like 100/1, with no sane person outside North America believing a victory was possible.

In 1775 US had no government, no army aside from some militia, no qualified general(aside from arguable Washington), modern weapons, navy, banks, credit unions, or allies to help them. The only thing American did have, which ultimately succeed their victory was determination for Independence, and desire for Freedom.

This is evident by almost every battle fought, as US needed to have far superior forces to push the British out of the field. In addition to this, US had a very hard tome keeping their soldiers in line, and equipped enough. Toward the end of the war, there were battles, were over 90% of all ammunition was provided by the French.

That all however, was almost insignificant to the war at sea. Wile US was very prosperous at the start of the war, their economy began to drastically decline. British would pick off US ships, causing the price of goods to raise extremely high, making it impossible to finance the war properly. US started to inflate their currency, which put them in a downward spiral.

It was only after the Battle of Saratoga, that France saw a possible victory should they, and all their allies enter the war against the isolated UK. France and Spain provided massive material, and more importantly naval aid to the US. UK had to keep most of their Navy in Europe, or France would steam roll them should they cross the channel, and therefore US could once again trade with the Caribbeans and other European nations. In addition, many European nations like Netherlands provided financial aid to the US, which helped the country further finance the war.

It is estimated that France provided between 1.5-1.7 Billion Livres, which would be about 1/3 of the entire US budget (Approximately $400 million which would equate to around 5 Billion Livres). That was France alone.

In conclusion, the war was not winnable, even if the British were pushed out of North America, as US economy would have collapsed, and the country would likely degrade into the dark ages similar to those in 700AD Europe.