Let $p\in [0,1]$, and consider a branching process where the number of offspring of an individual is zero with probability $p$, and is two with probability $1-p$. Initially there is one individual.
a) For what values of $p$ will the branching process become extinct after finite number of generations with probability 1?
b) For the case $p=1/4$, calculate the probability that the branching process survive forever.
Attempt:
Part a). Let me first compute the expected value of each generation. This would be $\mu=0*p+2(1-p)=2(1-p)$. For complete extinction, we need $\mu\leq1$. Using this requirement, we solve for $p$ to be inside $[0.5, 1]$.
Part b). let $x$=probability of complete extinction. Since we start with 1 individual, then $x=1*p+x^2(1-p)$. Since $p=1/4$, $x=1/3$.
Commnity wiki answer so the question can be marked as answered:
As Math1000 pointed out in a comment, your answers are correct.