Do it directly with the parent function, as shown in the figure (click to enlarge):

Question 2:

Here's the thing. It is possible to use the parent function, but it is more troublesome.

Since you say this is a question of probability theory, you should use some knowledge of probability theory to do it.

First of all, since this problem is given, we must know the "negative binomial distribution".

Actually, it doesn't matter if you don't know. The following statement does not require a negative binomial distribution.

The following statement comes from the conceited binomial distribution, which is essentially a negative binomial distribution.

As shown in the figure (click to enlarge):