Does a randomly chosen series diverge?
Solution 1:
For every $k$, $\mathbb E(P_k)=1/2^k$ thus: $$\mathbb E(S) = \mathbb E(P_1)+\mathbb E(P_2)+\cdots = 1/2 + 1/4 + \cdots= 1$$ Since $\mathbb E(S)$ is finite it follows that $P(S=\infty) = 0$, otherwise the expectation $\mathbb E(S)$ would be infinite.