In a town of $(n + 1)$ inhabitants, a person tells a rumor to a second person, who in turn repeats it to a third person, etc. At each step the recipient of the rumor is chosen at random from the $n$ people available.
(a) Find the probability that the rumor will be told $r$ times without:
(i.) returning to the originator;
(ii.) being repeated to any person.
Here's what I did:
(i.) For the first person, the probability is $1$ because they are the originator and can not tell to themselves. For the second, we can tell it to one of the $(n-1)$ people because they can't tell it to themselves nor to the originator, and so for the third. Thus: $1 * {(\frac{n-1}{n})}^{r-1}$. Note that considering that a person can not tell it to themselves then I devided the fraction by $n$ and not $n+1$.
(ii.) For the first person, again, we have probability of $1$. Then $n-1$ options for the second (not to themselves, and not to the orignator: $n+1 - 1 -1 = n-1$), then $n-2$ for the third (not to themselves, not to the first, and not to the previous: $n+1 - 1 - 1 -1 = n-2$. etc r times.
Is this good?
