Suppose that there are $n$ people alive in a population. Due to a deadly disease, each person dies with probability $\frac12$ each day (and there are no births). What is the probability that there will be exactly one person alive at some time?

Thoughts:

Let $p_k$ be the probability that the population reaches exactly $1$ person given that there are currently $k$ people alive. Then $p_0 = 0$ and $p_1 = 1$.

The probability of going from $k$ people alive to $k - j$ being alive (where $0 \leq j \leq k$) is the probability that $j$ die: $$ \binom{k}{j} \left(\frac{1}{2}\right)^j \left(\frac{1}{2}\right)^{k - j} = \binom{k}{j} \left(\frac{1}{2}\right)^k $$ And using conditional probability we have the recursion: $$ p_k = \frac{1}{2^k} \binom{k}{0} p_k + \frac{1}{2^k} \binom{k}{1} p_{k - 1} + \cdots + \frac{1}{2^k} \binom{k}{k - 1} p_1 + \frac{1}{2^k} \binom{k}{k} p_0, $$ or $$ (2^k - 1)p_k = \binom{k}{1} p_{k - 1} + \cdots + \binom{k}{k - 1} p_1. $$ Is it possible to solve a recursion like this? Is there a better way to solve the puzzle?