For the birthday problem, the probability that a set of n birthdays (where n=1) contains your birthday is 1/365. The complement is that there is a 364/365 chance that it does not contain your birthday.
P(B)=1/365
P(Bc)=364/365
P(B)=1 - 364/365
When you start adding birthdays to the set, it's valid to say that there is a (364/365)^n chance that your birthday will NOT be in the set. But for some reason you cannot say that there is a (1/365)^n chance that your birthday is in the set.
This is clearly true, otherwise the chance that your birthday was in the set would go to 0 as you added birthdays to the set. What I don't understand is how I am supposed to know ahead of time that multiplying the probabilities together works for the complement but not the non-complement.
Look up the inclusion-exclusion principle if you want to see how to compute the probability directly instead of the complement. But to answer your question perhaps a bit better, if you are dealing with the complement for your problem then the probability that each person does not have the same birthday as you are assumed statistically independent so you can just multiply to get the answer. However the probability that at least one person has the same birthday as you is not as simple, because if you have $k$ people the probability is at least $1/365$ but it's not $k/365$ because more than one person might have the same birthday as you, so there's overlap in cases. The inclusion-exclusion principle takes care of this. It's definitely not $(1/365)^k$, because that is the probability that all $k$ people have the same birthday as you.