Hi, My question is from Advanced Math/ Stats and Probability area. Problem: In a

Question

Hi,

My question is from Advanced Math/ Stats and Probability area.
Problem: In a town of 100 inhabitants, a person tells a rumor to a second person, who in turn tells it to a third person, etc. At each step, the recipient of the rumor is chosen at random from the people available. When someone tries to tell the rumor to someone who has already heard the rumor, they stop spreading the rumor (become a stiffler). Some people will not hear the rumor. What is the expected number of people to hear the rumor? Generalize to N size of the population.

In other words:
Total population: N+1 (1 - is the person who initially, at t=0, knows the rumor)

Definitions:
spreader (s) - person who knows the rumor and is actively telling it to other people
ignorant (i) - person who does not know about the rumor
stifler (r) - a spreader becomes a stifler when he meets someone who has already heard the rumor, which can be either another spreader or a stifler
So, the movement of people from one state to another is: S -> I -> R

There are 6 possible interactions:
spreader-spreader; spreader-ignorant; spreader-stifler; stifler-stifler; stifler-ignorant; ignorant-ignorant.
Only three are 'meaningful' (=change the state of a person):
1. Spreader-ignorant: ignorant becomes a spreader after getting in contact with the spreader
Ignorant subgroup is decreased by 1, Spreader subgroup is increased by 1.
2. Spreader-Stifler: the spreader becomes a stifler after encountering a person who already knows the rumor and thus losing interest in spreading the rumor
Spreader subgroup is decreased by 1, Stifler subgroup is increased by 1.
3. Spreader-spreader: when two spreader encounter each other, both become stiflers
Stifler subgroup is increased by 2, Spreader subgroup is decreased by 2.
Closed population, discrete time (i.e. period t=0,1,2,3,etc)
Possible ways to solve/ Keywords: Markov Process, Transition Matrix, random walk, probability, expected value, Daley-Kendall model

I have been reading a lot of advanced research related to this problem but cannot grasp how to start it: first solve and explicitly show a step-by-step analysis. Then, I need to generalize it to a population of N people. Please help with a detailed explanation and a step-by-step solution.

Thank you.

Explanation / Answer

I suggest you map it you. Based on probability, a spreader would do the most probably action. Thus keep a tally of how many people are spreaders and stiflers and ignorant. In the end keep in mind that there will be no spreaders left. If you need more help message me. Hope that helps!! :)

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