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Denote the survivors at time step 1 by \(S_n\). We wish to find \( \mathbb{P} (S_n = k ) \).

\[\begin{align*} \mathbb{P} ( S_n = k ) &= \sum_{i = k }^{\infty} e^{-\mu} \frac{\mu^{i}}{i!} \binom{i}{k} (p)^{i-k}(p)^k \\ &= e^{-\mu} \sum_{i = k }^{\infty } \frac{\mu^i}{i!} \frac{i!}{k! ( i - k )!} (p)^{i - k } (1-p)^k\\ &= e^{-\mu} \sum_{i=0}^{\infty} \frac{\mu^{ i + k} (p)^{i } (1-p)^k}{k! i!}\\ &= e^{-\mu} \sum_{i=0}^{\infty} \frac{ (p\mu)^i}{i!} \frac{((1-p)\mu)^k}{k!}\\ &= e^{-\mu} \frac{((1-p)\mu)^k}{k!} \sum_{i=0}^{\infty} \frac{ (p\mu)^i}{i!}\\ &= e^{-\mu} \frac{((1-p)\mu)^k}{k!} e^{p \mu }\\ &= e^{\mu ( 1 - p ) } \frac{((1-p)\mu)^k}{k!} \end{align*}\]