Probably chances of happening are high (or at least not minuscule) Possibly chances of happening are not zero
english.stackexchange.com/users/288/Šimon Tóth
These terms actually have fairly precise definitions, although they're sometimes used with a bit of liberty.
Let P = probability of event occurring Probably => 0.5 < P < 1
Possibly => 0.0 < P << 1
It is evident that in some cases, there is overlap between the two terms.
english.stackexchange.com/users/184/Noldorin
To say that something is probably the case is to say that that case is probable .
To say that something is possibly the case is to say that it is possible .
The two are quite distinct.
If something is probable, then it is likely to happen.
The chances are not only more than zero; they're better than 50/50 (or at least, the chances are higher that it'll happen than that it won't — since in comparing probabilities, you could have multiple items which have chances better than 50/50 but have one of them which is more probable than the others).
Probably indicates that the laws of probability say that it's more likely to happen than not to happen.
On the other hand, if something is possible, then it can happen.
Whether it will or not is irrelevant.
The chances are better than zero.
It doesn't say anything about how probable it is.
It could be that the chances are .000000001
% or that they're 100%.
By saying that something is possible, you're not really saying anything about probability.
You're just saying that it could — at least theoretically — happen.
So, if you say possibly , then you're saying that something could be the case, whereas if you're saying probably , you're saying that not only can
it be the case but that the chances are higher that it will happen than that it won't.
english.stackexchange.com/users/2170/Jonathan M Davis
Probably implies that the probability of an event is higher than that of the alternatives.
In the most common case with 2 alternatives, probably means something is more likely to happen than not.
With 5 alternatives, probability of the event is >20%.
Possibly implies a non-zero probability, usually a low one.
Conversational usage may not fit neatly into these definitions, as the intent is often mild exaggeration or irony.
english.stackexchange.com/users/1056/dbkk
Usage of Probably:
Probably, you should be able to do it. =
Mostly, you should be able to do it.
Usage of Possibly: Possibly, you should be able to do it.
[= If there is a chance (some factor implied), you should be able to do it.]
Possibly, you should be able to do it.
[= If it is possible, you should be able to do it.]
english.stackexchange.com/users/44409/arjun
I think the difference between probably and possibly is somewhat related to may and can .
E.g. Probably enough to make you laugh = may or may not be enough, not very sure about it, neither does it matter very much Possibly enough to make you sick = can be enough; warns of the ability (e.g. of a virus) to make you sick, and it's important to know
english.stackexchange.com/users/2044/mojuba
Wow... after reading the mumbo jumbo...
hahaha
I can't remember where I got those words.
Again, spanish helped me.
Probably =
Based on prudent reason - That can be proved - Said when there's good reasons to believe it will happen or will be verifiable.
Possible =
That can be or happen - that can be executed - available media to do something Possible
implies that exist the possibility, let's say 1%.
Probably implies that based on probability the chances to happen are good, let´s say above 50%.
Remember that Probable, probability, are math terms in statistics.
Possibility is not used because it works even against all odds.
Just having the minimal chance, something is possible.
But is probably not going to work, because the chances are too low.
english.stackexchange.com/users/2479/Billeeb
Recently I saw the movie "Pursuit of Happyness", which is actually quite good, and I noticed the actor (Will Smith) asking the difference between  probably  and  possibly  to his son. So I would like to have people's opinion on this question as I am still yet to find an answer myself.
What is the difference between "probably" and "possibly"?
english.stackexchange.com
<word-choice><differences><adverbs>
english.stackexchange.com/questions/6088
english.stackexchange.com/users/None/
"Probably" implies that the probability of the chances to happen are good, maybe above 50%. "Possibly", on the other hand, is used in situations where the probabilty of something happening are not zero. The terms have precise definitions although there is flexibility. In some cases, both terms overlap in meaning.Probable means "likely to happen" . "Possible" means it can happen. Probable means "likely to happen" . "Possible" means it can happen.Probably implies a higher likelihood of an even happening. Possibly means a lesser likelihood.The difference between them can be related to "may" and "can".
"Probably" implies that the probability of the chances to happen are good, maybe above 50%. "Possibly", on the other hand, is used in situations where the probabilty of something happening are not zero.  Probably  means the chances of happening are high. If something is probable, then it is likely to happen.  On the other hand, if something is possible, then it  can  happen and it  implies a non-zero probability, usually a low one.   
"Probably"  indicates that something is more likely to happen than not to happen. By saying that something is "possible", the probability is often low. "Possibly" merely suggests that something is possible. The key difference between these two terms is that "probably" is used when the chances of something happening are higher than those of it not happening whereas "possibly" is used in situations where the probabilty of something happening is simply greater than zero.
The terms have precise definitions although there is flexibility. In some cases, both terms overlap in meaning. Probable means "likely to happen" . "Possible" means it can happen. Probably implies a higher likelihood of an even happening. Possibly means a lesser likelihood. The difference between them can be related to "may" and "can".
 Probably  means the chances of happening are high. If something is probable, then it is likely to happen.  On the other hand, if something is possible, then it  can  happen and it  implies a non-zero probability, usually a low one.   
It you use the term "probably", you are saying that something is more likely to happen than not. If you use the term "possibly", you are saying that the likelihood of something happening is not zero.