What is the probability that the bacteria population eventually dies out? A jar begins with one bacteria. Every minute, every bacteria turns into 0,
1, 2, or 3 bacteria with a probability of 25% for each case (dies, does
nothing, splits into 2, or splits into 3). What is the probability that the
bacteria population eventually dies out?
 A: Let $p$ be the probability of the bacteria dying out(it also means tht none of its offspring remains alive),
So the probability of this bacteria dying out = either itself dies out or the bacterias generated by it also dies out
So we have,$p=\frac{1}{4}+\frac{1}{4}p+\frac{1}{4}p^2+\frac{1}{4}p^3$
(Reason: Cases may be either the bacteria itself dies out then there is no question of its offspring, or it remains as it is(with the probaility ($1/4$ ) and then it itself dies out (along with its next generation with the probability $p$ ,or it turns into 2 (with the probaility ($1/4$ )and then both of these diesout with its next generation with prob. $p^2$ and similarly for the case when it turns into 3 bacteria.) 
$\Rightarrow p^3+p^2-3p+1=0$
$\Rightarrow (p-1)(p^2+2p-1)=0$
$\Rightarrow (p^2+2p-1)=0$(As $p\ne1$ otherwise 'everything falls apart'.Truly, I cant see why it cant be 1, just my mind says so)
$\Rightarrow p=-1+\sqrt{2}$or $p=-1-\sqrt{2}$
As p cant be -ve so the answer is $p=\sqrt{2}-1$
