PMean: Will the baby walk away?

I submitted a cute probability problem to the Riddler section of Nate Silver’s Five Thirty Eight web site.  Here’s a cut-and-paste of the problem.

From Steve Simon, an adorable living-room puzzle:

Your baby is learning to walk. The baby begins by holding onto a couch. Whenever she is next to the couch, there is a 25 percent chance that she will take a step forward and a 75 percent chance that she will stay clutching the couch. If the baby is one or more steps away from the couch, there’s a 25 percent chance that she will take a step forward, a 25 percent chance she’ll stay in place and a 50 percent chance she’ll take one step back toward the couch.

In the long run, what percent of the time does the baby choose to clutch the couch?