Given that exactly 1 event happened within a group of 20 independent events, and knowing the probability of each event, how do I calculate the probability that a specific event happened?
Intuitively, I would say that it is $\frac{\text{Probability of Specific Event}}{\sum \text{Probability of each event}}$. Is this correct?
Let $\{E_i\}$ be your events.
Let $X_i$ be the event "The event $E_i$ happens but none of the events $E_j$ with $i\neq j$ happen". Then the independence of the $E_i$ tells us that $$P(X_i)=P(E_i)\times \prod_{j\neq i}(1-P(E_j))$$
Note that the $X_i$ are mutually exclusive, so the probability that exactly one of the $E_i$ happens is just $\sum P(X_i)$.
Your desired answer is then $$\boxed {\frac {P(X_i)}{\sum P(X_i)}}$$