Question

Consider a town with N people. A person spreads a rumor to a second, who in turn repeats it to a third, and so on. Suppose that at each stage, the recipient of the rumor is chosen at random from the remaining (N-1) people. Suppose rumor is repeated $n(\ge 3)$ times, then probability that it will not be repeated to

Answer 1

A,B,C) In this case the originator has (N-1) choices and each of the remaining (n-1) narrators has (N-2) choices. Thus, the number of favourable ways is $(N-1)(N-2{)}^{n-1}$. Hence, the probability of the required event is
$\frac{(N-1)(N-2{)}^{n-1}}{(N-1{)}^n}={\left(\frac{N-2}{N-1}\right)}^{n-1}={(1-\frac{1}{N-1})}^{n-1}$
Similar for B) and C)
D) Since at each stage the recipient is chosen at random from the remaining (N-1) people, the total number of ways is $(N-1{)}^n$. We now find the number of favourable ways. The originator has (N-1) choices, the second person has (N-2) choices (exclude the originator and the first recipient), the third person has (N-3) choices, ..., and the nth person has (N-n) choices. Thus, the number of favourable ways is (N-1) (N-2)...(N-n). Hence, the probability of the required event is
$\frac{(N-1)(N-2)...(N-n)}{(N-1{)}^n}=\frac{{}^{N-1}{P}_n}{(N-1{)}^n}$.