Probability - branching A population starts with 1 member: at t=1 , it can either divide with probability of p or dies with probability of 1-p. If it divides, then both of its children behave independently with the same options as above at t=2. What is the probability that no members in third generation? For what value of p is this probability equal to 0.5?
What I did:
P(NO MEMBER AT T=3)
=(1-p)+p(1-p)+p(1-p)
=-2p^2+p-1
I am not sure where is wrong, but p should be equal to 0.59697
Anyone can help?

Let $T_i$ denote the number of members in the $i$-th generation so that $P(T_2=0)=1-p$ and $P(T_2=2)=p$.
Then: $$P(T_3=0)=$$$$P(T_3=0\mid T_2=0)P(T_2=0)+P(T_3=0\mid T_2=2)P(T_2=2)=$$$$1(1-p)+(1-p)^2p$$
