1
$\begingroup$

Suppose that there are some bacterias. In any minute, each living dies with probability 1/4, stands still with probability 1/4, splits into 2 with probability 1/4, and splits into 3 with probability 1/4.

What is the probability of this species dying out finally, when initially there is only one bacteria?

  • 1
    Do you know about [probability generating functions](http://en.wikipedia.org/wiki/Probability_generating_function)?2012-02-02
  • 0
    @Chris Taylor: I am learning about it, thanks2012-02-02
  • 1
    Just a point of English: One bacterium, two or more bacteria.2012-02-02

1 Answers 1

6

The probability of eventual extinction of a branching process is the smallest root in $[0,1]$ of $\phi(t)=t$, where $\phi$ is the probability generating function. In your case, $\phi(t)=(1+t+t^2+t^3)/4$, and the probability is $\sqrt{2}-1=.41421$.

  • 0
    When I sum up over all timesteps (in my answer), I get $\sum_{t=0}^\infty (1/4)^{(3/2)^t}\approx 0.429407$, which is a little too large. Could it be used as upper bound?2012-02-02
  • 0
    @draks No, it's not that simple. I can find a different branching process with $p_0=1/4$ and $\mu=3/2$, but with arbitrarily large extinction probability.2012-02-02