Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 9G / a / Solution

Independence of individual bit errors gives $P=(1-p{)}^{4N}$. Independent retransmissions have a geometric distribution with success probability $P$, so the expected total number of transmissions, including the successful one, is $1/P$. At the supplied values, $P\approx 0.018279$ and $1/P\approx 54.71$.