This is assuming each trial has an independent probability.
In other words, lets say that I perform $50$ trials a $100$ times. I know that the event happened only in $5\%$ of those hundred $50$-trial sets. How can I estimate the probability of it happening in any given single trial?
Also, this probably has a well known name and solution. I would be happy to be pointed to it and reading more.
We assume that the only information that has been recorded is whether or not the event happened or did not happen in the various $50$-trial rounds. If we have more information, such as the total number $N$ of times the event happened in the $5000$ trials, then we make the natural estimate $N/5000$.
Let $p$ be the probability of the event happening in a single trial. Then the probability it does not happen in $50$ trials is $(1-p)^{50}$.
Our estimate for $(1-p)^{50}$ is $0.95$. So our estimate for $p$ is $1-(0.95)^{1/50}$.