Doubts on Mutually exclusive and Independent events

Problem:

In a school competition,the probability of hitting the target bu Dick is $\frac{1}{2}$,by Betty is $\frac{1}{3}$ and by Joe is $\frac{3}{5}$.If all of them fire independently,calculate the probability that the target will be hit.

This general approach for solving is to find the complement of the probability that the target would not be hit.

What I am not getting is that if the there events are mutually exclusive/independent then why not summing the three probabilities doesn't gives the correct answer?!

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The events are independent but not mutually exclusive. – Andrey Rekalo Dec 7 '10 at 11:30
If you sum the probabilities you obtain an answer greater than 1. This provides a quick sanity check that adding them is not the way to go. – Derek Jennings Dec 7 '10 at 11:38
"conditional probability" is not the correct phrase here. – Raphael Dec 7 '10 at 11:49
See my answer in math.stackexchange.com/questions/11415/…. – Shai Covo Dec 7 '10 at 12:22
More simply, try to understand why for any events $A$ and $B$, $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. – Shai Covo Dec 7 '10 at 13:13