I just thought of this problem after I read about how COVID trials have started. I drew inspiration from the 3blue1brown video on binomial distributions. Unfortunately that is all the context I have, so how I would I go about solving this problem with those principles in mind? I am relatively new in this field of mathematics so try to use terminology that a highschool senior who has BC calculus would understand. This is also my first Math Stack Exchange post so let me know if you need any more information.
Here is the 3blue1brown video I watched: https://www.youtube.com/watch?v=8idr1WZ1A7Q&vl=en
Researchers have found a COVID vaccine that causes harmful effects with probability $p$ where $p$ is uniformly distributed in the interval [0, 0.5]. To check the effectiveness of the vaccine, the researchers test the vaccine on 10000 volunteers and find that no one experiences adverse side effects. What is the smallest real number λ such that the researchers can assert p < λ with probability at least 90%?
the beta distribution with $a=1$ and $b=10001$ should give you your distribution of $p$
looks like $\lambda=2.5E-4$ puts you at 90% level