probability problem

This is in response to the following request:

A parent particle divides into 0,1,or 2 particles with probabilities 1/4,1/2,1/ disappears after splitting.let Xn denotes the number of particles in n-th generations with X0=1.find P(X2¿0) and the probabilities that X1=2 given that X2=1.;from=requests;id=927

For my first entry I will try to answer the question.

Let p0,p1 and p2 be the nonzero probabilities of dividing into 0, 1, or 2 particles, and let Xn denotes the number of particles at the nth generation.

With X0=1, find 1) P(X2>2) and 2) P(X1=2|X2=1)

1) After two generations there can be at most 22 particles so P(X2>2)=P(X2=3)+P(X2=4)


Note that if X2=3, then X2=2.


Using your values I get 3/32.

2) From the definition of conditional probabilityMathworldPlanetmath




Why? To get to X2=1, at n=1 there are either one or two particles, if there is one particle it remains one at n=2, and if there were two particles at n=1, then one has to go to zero and the other one—this can happen two ways.

Finally P(X1=1X2)=p1p2.


Using your values I get 2/3.

Now I have a question for you to think about. What happens in the long run, as n?

Title probability problem
Canonical name ProbabilityProblem
Date of creation 2013-03-22 19:11:21
Last modified on 2013-03-22 19:11:21
Owner statsCab (25915)
Last modified by statsCab (25915)
Numerical id 4
Author statsCab (25915)
Entry type Definition
Classification msc 62-01