just a quick question , what is the formula to determine how many "x" armies will get "wasted" and "y" armies "still remains" when :

x armies attack y armies
with z% luck.

for example 20 armies attack 25 armies ....
at 75 % luck
what will be remained at both places ?

as much i know the analyze button only tell us what should be the amount of armies to conquer the willing position .

There is no formula... the luck function makes it impossible to give an absolute answer but instead gives the most likely outcome (the analyze function).

Were it 0 luck, the attacking army (20) would kill 12 defenders (20*0.6) and the defending army (25) would kill 17-18 armies (25*0.7) thus leaving 13 defenders and 2-3 attackers. The higher the luck figure, the farther from these figures you will see.

The range of possible outcomes at 75% luck is between 3 (min 0 kills at 3/4 weight with expected value of 12) and 18 (max 20 kills at 3/4 weight with expected value of 12) defending armies killed and between 6 (min 0 kills at 3/4 weight with expected value 18) and 20 (max 25 at 3/4 weight with expected value 18) attacking armies killed.

The pseudo-random nature of the kills makes absolute predictions impossible.

You also need the offense and defense numbers