The Example:
There are 5 people in a room.
The chance of each one of them to get outside is $p$.
Then, the chance of a person outside being struck by a lightning is 0.05.
What is the chance that at least one person will be struck by lightning?
I was thinking about: $$ {5 \choose 1}p(1-p)^4*0.05 + {5 \choose 2}p^2(1-p)^3*(1-0.95^2) + ...+{5 \choose 5}p^5*(1-0.95^5) $$
Though it seems too long.
Is there a more efficient way to calculate this?