Verifiability is a key requirement for electronic voting. However, the use of cryptographic techniques to achieve it usually requires specialist knowledge to understand; hence voters cannot easily assess the validity of such arguments themselves. To address this, solutions have been proposed using simple tables and checks, which require only simple verification steps with almost no cryptography. This simplicity comes at a cost: numerous verification checks must be made on the tables to ensure their correctness, raising the question whether the success of all the small verification steps entails the overall goal of end-to-end verifiability while preserving vote secrecy. Do the final results reflect the voters' will? Moreover, do the verification steps leak information about the voters' choices? In this paper, we provide mathematical foundations and an associated methodology for defining and proving verifiability and voter privacy for table-based election protocols. We apply them to three case studies: the Eperio protocol, Scantegrity, and Chaum's Random-Sample Election protocol. Our methodology helps us, in all three cases, identify previously unknown problems that allow an election authority to cheat and modify the election outcome. Furthermore, it helps us formulate and verify the corrected versions.