For a multi-agent problem I want to calculate the probability that a certain event happens for any agent (1 or more). There are n agents, and $P(X_i=1)=p_i$ for each agent. I want to calculate the probability that the event is true for any agent: $p_c = P(X_1=1 \vee X_2=1 \vee ... \vee X_n=1)$.
$p_c = p_1 + p_2 - p_1 p_2$
And for $n=3$:
$p_c = p_1 + p_2 + p_3 - p_1 p_2 - p_1 p_3 - p_2 p_3 + p_1 p_2 p_3$
Is there an easier way to calculate this probability?