The policeman says that the fingerprints on the gun belong to a woman. It is now possible to tell the sex of the person who made a fingerprint but it was not when the movie was made.
When the Chans and the police discover that a suspect is on his way to the Chans' hotel with a duplicate key to their room, they drive there instead of telephoning, thus giving the thief time to enter.
While the police are searching Room 413, the telephone rings with a call for the inspector. Since the police went there straight from another room, rather than going past the front desk, there would be no way for the operator to know that the call should be put through to 413.