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I have a rather strange question (for this Stack Exchange anyway). It felt too mathematical to ask elsewhere. If this is out of place here, please let me know.

A chicken in Minecraft lays eggs; the time between layings is uniformly distributed between 5 and 10 minutes in intervals of 0.05 seconds. An egg, when thrown, produces one chick with probability $\frac3{32}$ or four chicks with probability $\frac1{32}$, and is destroyed afterwards regardless. A chick matures into an egg-laying chicken in 20 minutes. Assuming eggs are immediately thrown upon laying, how can I estimate the number of chickens after $X$ minutes starting with 1 chicken (that is at the start of its egg-laying cycle)? Chickens are immortal.

I can figure most of it out myself, but the thing that's giving me the most trouble is the last bit. I don't know how to take into account the time delay between an egg hatching and the chick growing up.