On a small island there are 25
inhabitants. One of these inhabitants,
named Jack, starts a rumor which
spreads around the isle. Any person
who hears the rumor continues
spreading it until he or she meets
someone who has heard the story
before. At that point, the person
stops spreading it, since nobody likes
to spread stale news.a) Do you think all 25 inhabitants will eventually hear the rumor or will the rumor die out before that happens? Estimate the proportion of inhabitants who will hear the rumor.
How would one go about calculating the probability? I would try to block it out, but there are so many potential branches. This problem is in the simulations portion of our statistics book, but I'm curious if there is any other way that doesn't involve simulating the situation.
Best Answer
Mike Spivey's remark that there are only three classes of people is quite relevant if you want to simulate this. At the start, there are 24 susceptible people and one infected. Call the number of each category S, I, and R. If two I meet, they both become R, so I goes down by 2 and R goes up by 2. If an S meets an I, the S becomes I, so S goes down by 1 and I goes up by 1. If R meets I, I goes down by 1 and R goes up by 1. Other meetings have no result. My guess is that somebody doesn't hear, because as S falls more and more meetings deplete the I's.