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There is a group of people, each individual with probability $0.2$ has $0$ child, with probability $0.6$ has $1$ child, with $0.2$ has $2$ children, then after a very long(you can think as infinity long) time

A This group will be extinct
B This group will have a population of infinity
C The population of this group will be stable in a range
D Not enough information

My solution:
Intuitively, Because Expectation=$0.2*0+0.6*1+0.2*0.2=1$, so I feel it should be stable in a range. But why this is wrong?

I want to get a feeling why it will be extinct? Or how can we explain it intuitively?

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Doesn't each person have two parents, making the population half in size each generation, and eventually exctinct? –  Jan Dvorak Mar 23 '13 at 22:35
    
how did you get to the "stable in a range"? if everyone has one child (in average), thus there is infinity people of infinitely generations. (But I don't understand the sense of the question and answers) –  V-X Mar 23 '13 at 22:35
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This is a special case of branching process, look out this question for some insight: math.stackexchange.com/questions/295353/branching-process –  Jean-Sébastien Mar 23 '13 at 22:35
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One can show that almost surely, there will be extinction of the population if the mean of the reproduction distribution is less than or equal to $1$, provided $0$ child is possible. Unfortunately i can't type the proof, but it has to do with fixed point of the generating function of the reproduction law. See for example cims.nyu.edu/~csplash/data/notes/2011-24.pdf –  Jean-Sébastien Mar 23 '13 at 22:38

1 Answer 1

This is a case of Gambler's ruin. If you keep trying long enough, with probability $1$ the group will go extinct. As written, the question does not specify how many generations you try for, so you might have a good chance of still having some individuals.

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After a very long time –  nkhuyu Mar 23 '13 at 22:40
    
@nkhuyu $O(pop^2)$ –  Jan Dvorak Mar 23 '13 at 22:41
    
as opposed to $(\ln(pop))$ if the expectation is below $1$ –  Jan Dvorak Mar 23 '13 at 22:42

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