Advantages and possible usages of encryption schemes with probabilistic decryption
Since when decrypting we always want to get the correct message back, there's no reason why we would want to make this ambiguous. It would have no security advantage (if the adversary can guess with any non-negligible probability, you have already lost, so ambiguous decryption can't make that harder). Thus, unlike probabilistic encryption, which is needed for security, probabilistic decryption is an undesired property that sometimes arises when trying to construct schemes. For example, with the naive Rabin, it just happens to be a property of the number theory used. The additional work required to make Rabin a trapdoor permutation is necessary to make it useful.
Thus, the only reason why randomized decryption (with a probability of error) exists is when this is what some constructions unfortunately give us. (This can be viewed in some sense as an advantage - if we weren't able to allow decryption errors, we wouldn't be able to build these schemes, some of them having advantages.) There are some examples of this: NTRU and the Ajtai scheme based on lattices. As a result, there has been some theoretical work on removing errors from encryption schemes that have them. Most notably, I recommend reading the paper Immunizing Encryption Schemes from Decryption Errors by Dwork, Naor and Reingold.