From: israel@math.ubc.ca (Robert Israel)
Newsgroups: sci.math
Subject: Re: Infinite case probability
Date: 3 Sep 1998 23:28:13 GMT

In article <1998090221305600.RAA26004@ladder01.news.aol.com>, lporter33@aol.com (LPorter33) writes:
|> <<A single alien from outter space has landed at Roswell, New Mexico.  His life
|> cycle and reproductive system is entirely different from those of animals on
|> earth.  In the earth environment, each day there is one chance in seven that he
|> will die, four chances in seven that he will live but not reproduce that day,
|> and two chances out of seven that he will reproduce.  When he reproduces, he is
|> almost immediately replaced by two mature copies of himself (rather like an
|> amoeba dividing) which will then independently die, live, and reproduce with
|> the same probabilities on the next day.  What is the overall probability that
|> this alien species will become extinct on earth?>>

|>  This problem comes from a math contest board. I have trouble with these types
|> of infinite, repetitive cases. I usually try a standard tree diagram, but it
|> doesn't come out right. If anyone knows a way of solving this problem, or these
|> types of probability problems in general I would greatly appreciate it.

Jose Nieto posted the solution, but perhaps it should be mentioned that this
is an example of what probabilists call a "branching process".  For more
information on these, you might look at Ross, "Introduction to Probability
Models", Taylor and Karlin, "An Introduction to Stochastic Modeling", or the
old classic Feller, "An Introduction to Probability Theory and Its Applications".

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2