The Self-Replicating Probe

A deep-space probe is designed to build copies of itself before shutting down: it produces 0 copies with probability $\frac{1}{4}$, 1 copy with probability $\frac{1}{4}$, or 2 copies with probability $\frac{1}{2}$, and every copy later does the same, independently. Starting from a single probe, what is the probability that the entire line of probes eventually dies out?