Probability of multiple events: continued I have 5 independant events.  How do I calculate the probability of event A occurring + 1 other?  How do I calculate the probability of event A occurring + 2 others?  What about the occurrence of A & B plus one other, but not the rest?  I have multiple variations of this I'm trying to figure out.
Event A:  15%
Event B:  30%
Event C:  15%
Event D:  10%
Event E:  10%

The key here is to leverage independence. The first thing to note is that $A$ is independent of the event "at least one of $B,C,D,E$ occurs", so that
$$
P(A\cap\{\text{at least one of }B,C,D,E\})=P(A)\cdot P(\text{at least one of }B,C,D,E).
$$
So, how can you compute the probability that at least one of $B,C,D,E$ occurs? Well, for starters,
$$
P(\text{at least one of }B,C,D,E)=1-P(\bar{B}\cap\bar{C}\cap\bar{D}\cap\bar{E}),
$$
where $\bar{X}$ denotes the event that $X$ does not occur. But this probability is not bad to compute: the fact that $B,C,D,E$ are independent implies that $\bar{B},\bar{C},\bar{D},\bar{E}$ are independent as well, so that
$$
\begin{align*}
P(\bar{B}\cap\bar{C}\cap\bar{D}\cap\bar{E})&=P(\bar{B})\cdot P(\bar{C})\cdot P(\bar{D})\cdot P(\bar{E})\\
&=0.7\cdot0.85\cdot0.9\cdot0.9\\
&=0.48195.
\end{align*}
$$
Thus the probability that at least one of the events occurs is
$$
P(\text{at least one of }B,C,D,E)=1-0.48195=0.51805.
$$
I'll let you finish it up from here.
