TITLE: What is the probability that you never lose this hypothetical dice game?
QUESTION [5 upvotes]: You have a four sided die numbered 1-4 and are playing a game. On the first round, you roll the die once. If you get a 1 you lose the die. If you get a  2 you keep the die. If you get a 3 or 4 you get another, identical die. On the second round you roll each die that you have, and the same thing happens with each die. Once you finish rolling all your dice you move on to the third round and so on. If you have zero dice you lose the game. What is the probability that you never lose the game over infinite rounds?

REPLY [5 votes]: This is a classic example of a branching process.  As other answers have pointed out, the extinction prob $p = Prob(lose) = 1$ or $1/2$.  To pick the correct answer, the linked wikipedia article has this to say:

it can be shown that starting with one individual in generation zero, the expected size of generation $n$ equals $\mu^n$ where $\mu$ is the expected number of children of each individual. If $\mu < 1$, then the expected number of individuals goes rapidly to zero, which implies ultimate extinction with probability $1$ by Markov's inequality. Alternatively, if $\mu > 1$, then the probability of ultimate extinction is less than $1$.

In the OP example, $\mu = (0 + 1 + 2 + 2) / 4 = 5/4 > 1$, so if you believe the wikipedia article, then $p < 1$.  So the right answer is:
$$p = \frac12$$
Disclaimer: I personally don't know enough about branching processes to verify what wikipedia said.  The article has some more details (in addition to what's quoted above) but I haven't read it in detail.