Spread of rumors. 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 and so on. At each step the recipient of the rumor is chosen at random from the n people available. I want to find the probability that rumor will be told r times without A)Returning to originator and
B)without being repeated to any person.
Secondly, How to solve this question when at each step the rumor is told to gathering of N randomly chosen people.
Let $R$ be the event that the rumor is told $r$ times without being told to any person who already heard it. We can say right away that $P(R|r \ge n)=0$, since there are only $n+1$ people.
Now consider this. For the first person, there are $n$ people to tell that have not yet heard it. Thus the probability of not repeating it to someone who knows it is $\frac{n}{n}=1$. For the second person, there are $n$ people to tell and $n-1$ do not know it already, so the probability for that is $\frac{n-1}{n}$. Basically we have that, after $n$ "tells", the probability is $$\frac{n}{n}*\frac{n-1}{n}*\frac{n-2}{n}*...*\frac{n-r+1}{n}$$ which is the same as $$\frac{n!}{(n-r)!n^r}$$ and that should be the answer.
