Probability doubt

[MR]

In an hour, a bacteria dies with probability p or else splits into two. What is the probability that a single bacteria produces a population that will never become extinct?

[cx]

2/3

[MR]

No. The answer is 1-2p/1-p.

[MR]

But I don't know how to arrive to it.

[Sinistral]

let t be the probability that bacteria goes extinct.
so t = p + (1-p)t^2  (where first term indicates the very first bacteria dies and hence no further population. 2nd term indicates 1st bacteria splits AND the 2 offspring get extinct -- probability of extinction is t )

(1-p)t^2 - t + p =0
quadratic in t

t= {1 ± sqrt[ 1 - 4p(1-p)] }/2(1-p)
  = 1, p/(1-p)

so Pr(that doesnt go extinct)= 1 - [p/(1-p)]
                                       = (1-2p)/(1-p)