What is the ruling on prayer when in doubt about the transfer of ritual impurity from a wet hand to ceramic, and when unaware of the location of the impurity on a prayer rug?

The water falling from the hand is pure unless it separated from an impurity before the impurity was removed. If there is doubt about its contact with impurity, the original ruling is purity. As for a ritually impure rug, it is not permissible to pray on it before purifying it. If the location of the impurity is hidden and cannot be pinpointed, the entire rug must be washed. All of this applies when one is certain of the presence of impurity; however, if there is doubt, the original ruling is purity.