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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.

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@CrazyJugglerGrummer: Then it should be tagged as homework... which I did for you. –  Arturo Magidin Nov 7 '10 at 2:50
    
Take a look at the logistic function. –  a.r. Nov 7 '10 at 3:00

2 Answers 2

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.

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This description also gives a dynamic programming algorithm (q.v.) that can find the exact probability in $O(n^2)$, where $n = 25$ in our case. –  Yuval Filmus Nov 7 '10 at 5:52

An SIR model is also appropriate because of this aspect of the problem: "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." Here, susceptible = hasn't heard rumor, infected = has heard rumor but is still willing to spread it, and recovered = has heard rumor but is no longer willing to spread it.

I see you're in 9th grade, and so the differential equations in the SIR model may not be something you've seen yet. Still, you may get something out of reading the explanation of the model on the Wikipedia page. Also, the standard SIR model doesn't quite fit this scenario; the equations for the infected and recovered populations for your scenario should be

$$\frac{dI}{dt} = \beta SI - \gamma RI,$$ $$\frac{dR}{dt} = \gamma RI.$$

I know you said you only wanted hints, but I'm giving you a fuller explanation of the SIR model because, according to this paper, "While the formal model is simple, in general it cannot be solved analytically." So this may just be one of those problems for which simulation is the best solution we have now.

O.K., I have to add this: For fun, you should also check out the SIRZ (susceptible-infected-recovered-zombie) model. We can all be thankful that the authors say, "Most of the analysis concluded that, under reasonable circumstances, a zombie apocalypse is unlikely."

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