Could anyone help me? A bacteria has 1/3 probability to die and 2/3 probability to divide into 2 identical cells. What is the probability that all the bacteria die ? That is: finally, there is no bacteria anymore.

Are you looking for the prob that all will all die over an n generations or just after the first generation?

If there are P bacteria in the initial (0th generation) population then the prob they all die after the 0th generation is

(1/3)(1/3)(1/3)(1/3) ...(1/3) with P factors.

=(1/3)^P

It gets more complicated if you want to extend to n generations.