Interpretation of the Negative Binomial and Geometric distributions

Question

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?

Answer 1

Extended Comment:

I think it would be helpful for you to use more specific language: (a) When you say "the probability that I achieve 5 successes within 10 trials," you need to say 'exactly 5' or 'at least 5'. Even if it is clear to you what you mean, it is not necessarily clear to others. (b) Also, you need to distinguish between what you 'control' for the experiment and what you 'specify' for your event.

$Binomial.$ You control the number of trials $n = 10$. And you control the success probability $\theta = 0.3.$ Then $X_{10} \sim Binom(n, \theta)$ records the number of Successes obtained. You may specify the event $\{X = 5\}$ or $\{X \ge 5\}$.

Then $P(X_{10} = 5)= 0.1029$ and $P(X_{10} \ge 5) = 0.1503$, according to the following computations in R:

 dbinom(5, 10, .3);  sum(dbinom(5:10, 10, .3))
 ## 0.1029193
 ## 0.1502683

For $n = 30,$ you have $P(X_{30} = 5)= 0.0464 < 0.1029$ (as you observed) and $P(X_{30} \ge 5) = 0.9698 > 0.1503$ (as @AndreNicolas commented):

 dbinom(5, 30, .3);  sum(dbinom(5:30, 30, .3))
 ## 0.04643981
 ## 0.969845

I think your problem is more directly handled in terms of binomial probabilities than negative binomial (or geometric) ones. You don't care about the arrangement of successes and failures along the sequence of 10 or 30 trials. In particular, you don't care whether the last trial is a success.