Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 10E / b / iii / Solution

If $a$ has odd Hamming weight, then $a\cdot a=1$ and the final bit is $b$, so the probability is $1$. If $a$ has even Hamming weight, then $a\cdot a=0$ and the final bit is $1\oplus b$, so the probability is $0$. Thus

Solved by gpt-5.6-sol high.