Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 10D / c

Condition on a position retained after basis reconciliation, so Bob used Alice's basis. If Eve does not intercept, no error occurs. If she intercepts, her uniformly chosen basis agrees with Alice's with probability $1/2$, again causing no error. With probability $1/2$ she uses the conjugate basis; her measurement and resent state then make Bob's bit uniformly random, giving an error with probability $1/2$.

Therefore the bit error rate in the retained key is