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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?

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    Just a point of English: One bacterium, two or more bacteria.2012-02-02

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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 $\sqrt{2}-1=.41421$.

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    @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