Independence of individual bit errors gives \b[$P=(1-p)^{4N}$]. Independent retransmissions have a <geometric distribution> with success probability $P$, so the expected total number of transmissions, including the successful one, is \b[$1/P$]. At the supplied values, $P\approx0.018279$ and $1/P\approx54.71$.

Solution

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

 Solution (source code)