Using simple probability concept,
Suppose we take a n-bit string. Now looking into the sample space, if we want the same string again, we only have $1$ possibility.
Now, sample space = $2^n$ [combinations of n-bit strings]
Probability to get same identical string = $\frac{1}{2^n}$
But we want the opposite.
Therefore, $1-\frac{1}{2^n}$
This returns the probability of atleast one character that will be different from the original string.