John and Mary both pick passwords at random from a $k$ letter alphabet. There are up to $n$ letters allowed in the password. Repetitions are allowed. What is the probability that they pick the same password?
There are $\sum _{i=1}^{n}k^i$ possible passwords, and $\frac{1}{\sum _{i=1}^{n}k^i}$ probability of picking any particular password.
I'm not sure how to set the problem up from here. I know that the answer is either John or Mary picks a password and then the probability of the other picking the same password is $\frac{1}{\sum _{i=1}^{n}k^i}$, but I'm not sure why. I initially thought it would be $(\frac{1}{\sum _{i=1}^{n}k^i})^2$