Question

A dormitory has n students, all of whom like to gossip. one of the students hears a rumor, and tells it to one of the other n − 1 students picked at random. subsequently, each student who hears the rumor tells it to a student picked at random from the dormitory (excluding, of course, himself/herself and the person from whom he/she heard the rumor). let pr be the probability that the rumor is told r times without coming back to a student who has already

Answer 1

Answer:

$P(r)=\frac{(n-3)(n-4)....(n-r)}{(n-2)^{r-2}}$

Step-by-step explanation:

From the question, we have the following condition:

$p_1=p_2=1,\:and\:p_n=0$

We know that each student who hears the rumor tells it to a student picked at random from the dormitory (excluding, of course, himself/herself and the person from whom he/she heard the rumor)

The 3rd student can therefore tell the rumour to n-2 students but only n-3 will accept it.

$\implies p(3)=\frac{n-3}{n-2}$

Consequently, the 4th student must not tell the 1st and second students.

$\implies p(4)=\frac{n-3}{n-2}\times \frac{n-4}{n-2}$

We can rewrite this to observe a pattern:

$\implies p(4)=\frac{(n-3)(n-4)}{(n-2)^2}$

$\implies p(4)=\frac{(n-3)(n-4)}{(n-2)^{4-2}}$

$\implies p(r)=\frac{(n-3)(n-4)(n-5)...(n-r)}{(n-2)^{r-2}}$

Hence, the probability that the rumor is told r times without coming back to a student who has already is:

$P(r)=\frac{(n-3)(n-4)....(n-r)}{(n-2)^{r-2}}$

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