As stated in the Risk FAQ the expected losses per attack for standard Risk rules is about 6 to 7. This means the attacker is expected to lose 6 armies for every 7 defender armies destroyed. Since we're talking about expected values, this represents the mean (average), which is most akin to a 50th percentile or 50% chance that that is what will happen. (this is not techically correct in terms of statistics, but I'm trying to explain it in more layman's terms).
As for 'chances to win a battle' this is very difficult to produce a rule of thumb for, since the numbers vary wildly depending on # of armies in play. For example, an 'even match' of 100 to 100 is won with over 85% chance, but 10 to 10 is only about 50%-50%.
It would be easier if you specified a particular odds at which you want to attack or not, then a simple matrix of attacker/defender army counts is possible to create (perhaps with an easy formula), but when you start with N attackers and M defenders the best 'answer', statistically, is a probablilty distribution function which isn't easy to calculate.
Assuming you want to attack at, at least 50% expected value (as above), use the 6 to 7 rule. This will also tell you how many you are expected to have left, so, using this rule, if you have 20 armies (to attack with, so 21 in the country), and he has 21, the 6 to 7 rule would say that you're expected to lose 18, killing his 21 (6*3, 7*3), thus leaving 2 left over.