I am having trouble putting together the way these distributions work. It doesn't matter whether we speak of the support space in terms of number of trials or failures.
Specifically, what variable is it we have control over?
I ask this because, say, we want to find out the probability that I achieve $5$ successes within $10$ trials compared to obtaining $5$ successes within $30$ trials with probability of success being $0.3$ and of failure $0.7$. Numerically I personally would've thought that the probability would have gone up with the larger number of trials, but instead it went down.
Do I control only the number of trials that occur and then whatever the different amounts of successes that occur from said event the i take those as is or do can i also state the amount of successes that I want within a certain range?