Probability of B and C happening at least once, in five independent trials. I am performing a product trial. The trial has three possible outcomes with the following probabilities: 

Outcome A = .3

Outcome B = .5

Outcome C = .2


If I perform five trials, what are the odds of B and C occurring at least once throughout testing?
The complement of this event is that only $A$ and $C$ occur or only $A$ and $B$ occur. The probabilities for those are $(.3+.2)^5=.5^5$ and $(.3+.5)^5=.8^5$, respectively. However, adding these two double-counts the possibility that only $A$ occurs, so we have to subtract $.3^5$, for a total of

$$
.5^5+.8^5-.3^5=0.3565\;.
$$

Thus the probability of your event is $1-0.3565=0.6435$.
This is the complement of the probability that all five are outcome A.  Assuming independence this complement is 1 - .3^5 = .99757

Sorry this is for B or C occurring at least once.  Please downvote more.