I'm wondering how to solve a questions of the example form:
How many dice are needed to make rolling at least 3 sixes in a single throw probable (p>0.5)
I know how to solve the question by graphing out all of the binomial probabilities for successive numbers of dice (for $n\geq3$ dice, compute $P_n(\geq 3$ sixes$) = 1 - P_n(0$ sixes$) - P_n(1$ six$) - P_n(2$ sixes$)$ each time and find the least $n$ such that $P_n(\geq 3$ sixes$) \geq 0.5$).
Enumerating out the probabilities for different numbers of dice, the answer I get is 16.
But one could of course consider generalising the question to arbitrary-sided dice, different number of successes, different target probability, etc. I'm wondering if there a more direct method to solve the general form of such a problem?
(There are already lots of similar questions on here, but the ones I found tackle the more classical question of computing the probability of a specific number of successes for a specific number of trials, not the number of trials required to make [at least] a fixed number of successes probable.)