Final answer:
The correct word to describe investments that were profitable by chance is 'fortuitous'. It indicates that the success occurred accidentally, not due to informed planning.
Step-by-step explanation:
The student's question asks for the most appropriate word to describe investments that turned out to be profitable by chance, rather than through informed decision-making. From the options provided, the correct word to fill in the blank is fortuitous, which means happening by accident or chance rather than through intention or planning. Therefore, the complete sentence would be: 'His investments proved to be profitable, but they were fortuitous rather than the result of knowledge and planning.'