I am given the question: A viral meme starts in one account, and is re-shared by $\mathbf{M}$ other accounts, where $\mathbf{M}$ is a non-negative discrete random variable. Assume that the future behaviour following from each of the initial re-shares is independent and distributed identically to the sharing behaviour from the initial creation: after $\mathbf{m}$ initial re-shares we have $\mathbf{m}$ independent and identically distributed copies of the initial meme process. Let $\mathbf{s}$ be the probability that the meme eventually dies out, which is also the probability that any of the $\mathbf{m}$ sub-branches of the process dies out. We define s as: $$ s = \sum_{m = 0}^\infty P(M=m)s^m = E(s^M)$$ and using the fact that now M $\sim$ Bin(3,1/2). Calculate $M_{M}(t)$ and hence find plausible values for s.
So I think the $M_{M}t$ is just $(0.5+0.5e^t)^3$ and now to find plausible values of s. I was thinking that using the fact that M is now a Binomial random variable. We can use this to substitue the pmf of M in the sum.As in, $$ s = \sum_{m = 0}^3 \binom{3}{m} 0.5^m 0.5^{3-m}s^m$$ And this would produce some cubic which we can solve for s.