probability (spread a rumor) In a room with $n+1$ people, a person tells a rumor to another person, who in turn repeats it to a third person, and the process continues. at each step the receipt of the rumor is randomly chosen out of the n other people in the room. 
a). Find the probability that the rumor reaches the originator of the rumor in exactly $r$ steps.
b). Find the expected number of steps for the rumor to reach the originator.
I answered the part a as following: $p(\text{the originator})= (n-1/n)^{r-1}$. 
I have no idea about the part b. Can anyone help me out about the solutions and explain the concept? 
What kind of materials that I need to study with because my professor never covers these materials in the class. Thank you very much.
 A: For each step, the probability to pass the rumor to the originator is $1/n$ while the probability to pass the rumor to someone else is $(n-1)/n$.
Assuming $r=2$ corresponds to the case that the first person repeats it to the originator, then for there to be r steps, simply multiply the corresponding probabilities,
$$P(r)=\left({n-1\over n}\right)^{r-2}{1\over n}$$
With this you can calculate $E[R] =n+1$. You may have a different definition for $r$.
A: HINT:
Let $X$ denote the random variable representing the number of steps it takes for the rumour to reach the originator.
$$\Bbb E[X]=\sum_{r=1}^\infty r\cdot\Bbb P(X=r)$$
EDIT: Requested by OP to expand on this
As stated in the other answer, $$P(X=r)=\left(\frac{n-1}n\right)^{r-2}\frac1n$$ for $r\ge2$. Thus $$\begin{align}E[X]&=\sum_{r=2}^\infty\frac1n\left(\frac{n-1}n\right)^{r-2}\cdot r\\
&=\frac1n\sum_{r=2}^\infty\left[\frac{d}{dx}\left(x^{r-1}\right)-x^{r-2}\right]\end{align}$$ where $x=\frac{n-1}n$.
$$\begin{align}nE[X]&=\frac{d}{dx}\left[\sum_{r=1}^\infty x^{r}\right]-\sum_{r=0}^\infty x^r\\
&=\frac d{dx}\left(\frac{x}{1-x}\right)-\frac1{1-x}\\
&=\frac{1}{1-x}+\frac{x}{(1-x)^2}-\frac1{1-x}\\
&=\frac{\frac{n-1}n}{\left(\frac1n\right)^2}\\
&=n(n-1)\end{align}$$
Thus we conclude that $$\bbox[5px,border:2px solid red]{E[X]=n-1}$$
