In an unusual arrangement with a commercial company, dozens of public universities plan to offer an introductory online course free and for credit to anyone worldwide, in the hope that those who pass will pay tuition to complete a degree program. The universities — including Arizona State, the University of Cincinnati and the University of Arkansas system — will choose which of their existing online courses to convert to a massive open online course, or MOOC, in the new program, called MOOC2Degree. The proliferation of free online courses from top universities like Harvard and Stanford over the past year has prompted great interest in online learning. But those courses, so far, have generally not carried credit. “We’re taking the MOOC idea, but now it will be part of a degree program, not a novelty,” said Randy Best, the chairman of Academic Partnerships, a company that helps public universities move their courses online.