I am new to probability , I do one question and stuck for days. Would you all can give me a hint on it?

There are (n \geq 1) balls in a bin. n-1 are black, and 1 is red. (k\geq n) players each draw a ball from the bin randomly in a round-robin fashion, until the red ball is drawn.

a) What is the probability that the i th player draw the red ball, for i\in \{1,2,,k\}?

b) Suppose there are 2 players,r \geq 1 red balls and (n-r) black balls in the bin, what is the probability that first player draw the red ball first?

I just think about the result of two players game,

P(A)=A+A'B'A+A'B'A'B'A+\dots

P(A)=\frac{r}{n}+\frac{r}{n-2}\cdot\frac{n-r}{n}\cdot\frac{n-r-1}{n-1}+\frac{r}{n-4}\cdot\frac{n-r-1}{n-1}\cdot\frac{n-r-2}{n-2}\cdot\frac{n-r-3}{n-3}\cdot\frac{n-r-4}{n-4}+\dots

am I doing in the right way or not, if yes, how should I compute this sequence?


c) Suppose the balls are drawn with replacement instead, what are the answers to a) and b)?


c)b) P(A)=A+A'B'A+A'B'A'B'A+\dots

P(A)=\frac{r}{n}+\frac{r}{n}\cdot\frac{n-r}{n}\cdot\frac{n-r}{n}+\frac{r}{n}\cdot\frac{n-r}{n}\cdot\frac{n-r}{n}\cdot\frac{n-r}{n}\cdot\frac{n-r}{n}+\dots


P(A)=\frac{r}{n}\cdot\frac{1}{1-(\frac{n-r}{n})^2}

Am I doing in the correct way?


And about a) and c)a)
I fail when the number of i is different, I don't really know what exactly it is running. Can anyone give me a hint on it?

Thank so much