By solving the lottery problem, it is found that the reason leading to different probability results lies in whether there is a return operation.
For the problem of key opening, if the incorrect key that has been tried is not considered in the next attempt to open the door, then this is a lottery (lottery) problem with no return, and the result should be1/n; Otherwise, the result is ((n-1)/n) (k-1)) * (1/n).