Let $a ,$ $b ,$ $\alpha ,$ and $\beta$ be non-zero binary strings of length exactly $n$ where $ \beta = a \alpha + b. $
Now, consider the following scenario. Fix $a$ and $b.$ I give you $\alpha$ and $\beta,$ but keep $a$ and $b$ as secrets from you. What is the probability that you can modify both strings (same length, just alter some bits), such that the altered strings \alpha' and \beta' still satisfy \beta' = a \alpha' + b. A cryptography paper which I am reading mentions it is $2^{-n}$ but I couldn't convince myself why.