Biden is easily the third most likely person to be the next President. It's the electoral college tie + hung House scenario.
You realize that in the event of a tie where the Senate elects the Vice President they are not obligated to vote for one of the candidates. They can choose anyone. In fact, in the event of an actual deadlock with no President, I doubt they would pick anyone from either ticket at the risk of looking too partisan, and might well settle for someone like Feinstein who would be at least minimally tolerable to Republicans. While the Democrats could conceivably push through Biden, doing so after they failed to ram through Obama in the house(and it would only go to the Senate if the Democrats failed to hold their caucus together in the house) would look much worse than trying to get a unanimous vote for interim President.
