Probability of impossible event. There is question in my book: Probability of impossible event is?

After reading the question my instant answer was $0$ and that was the answer given.

But then i thought other way, question is probability of impossible event, so there are two outcomes possible or impossible (event can be certain or impossible).

Therefore probability of impossible outcome is $\frac{1}{2}$.

Can that also be answer?
Your first answer is right. Your second argument only works to compute the probabilities of equally likely outcomes. So it's perfectly good to figure out the chances of heads coming up from a coin flip. But if one outcome is specified to be impossible, then by definition you are not working with equally likely outcomes and so the computation breaks down.
It depends what you mean by "Probability of impossible event".  This is an ambiguous phrase which would be interpreted in several different ways:

1) The event is known ahead of time to be not possible, therefore by definition in mathematics, the probability is defined to be 0 which means it can never happen.  I suspect most people on this site will go for this answer.  An example would be drawing a joker from a standard (thus jokerless) $52$ card deck of playing cards.  It aint gunna happen.

2) It is not known ahead of time if an event is possible or impossible and you are asking what are the chances it may be impossible.  For example, flipping a coin so it lands on its edge.  Some people think this is impossible since they seem to always say $50$% chance heads and $50$% tails but that is wrong cuz it depends where it "lands".  I've seen a coin land on its side and stay there.  What if you flip it and it lands in soft sand for example?  

3) A certain event may seem impossible now but in the future it may be possible.  For example, it was thought to be impossible that a dirt bike (motorcycle) could ever do a double backflip ramp to ramp in competition but it has been done fairly recently (2006 X-games by Travis Pastrana).  Actually to me it seems more like a double reverse flip (facing forwards but flipping backwards).  Possible outcomes for this event would be success without injury, success but with injury, failure with injury, death....

Is there a time constraint on your event?  If not, then it may just take a while for the seemingly impossible event to happen and thus confirm it is possible.

So my answer is depending on what definition/interpretation of "impossible" you use, the probability can differ although I wouldn't know how to compute any interpretation other than the interpretation of truly not possible.