I need help with the following problem:

Problem 2. Branching process to model rumor spread (20 pts) We consider a branching process (Zn)neN given by the following dynamics: 0 With probability 1 3), an individual has 0 offspring. a With probability p, the individual has a positive number of offspring decided as described below. a The individual performs an experiment with failure probability q and success probability 1 q, repeatedly until a success occurs. The number of offspring is given by the number of trials required. will decide to spread it or not with probability 1/2. If they decide to spread it, every time they tell the rumor to someone there is a probability 1/3 that they'll decide not to spread it further. One person starts a rumor. What is the probability that the rumor will eventually die out ? (e) What is the expected number of people who have heard the rumor after 3 steps of spread