A simple request for an answer

I am no mathmetician but perhaps someone could help answer a question
from an old fool but here is a question about mapping the galaxy using
self-replicating robots.

Assuming we send 1 probe initally:


Given the probe travels at half the speed of light in a fixed direction
and has a 5% chance per-light year of running into a planet that it can
use to replicate (assume it is successful the first time as we would
aim it at something) and it takes 'A' years to build the necessary
tools needed to launch new probes and launches 'B' number of probes in
opposing directions with a 'C' failure rate of new probes. How long and
how many probes need to be built and launched to map the galaxy?


I assume that failure rates would be less then 50% on replicated probes.