Baby Couch Walk

Baby Couch Walk
ProbabilityEasy

Baby Diyu is learning to walk with the support of a couch in the living room. She starts at the couch and makes a decision at each moment. She will either bravely take a step forward with a probability of 0.2, or stand still not knowing what to do with a probability of 0.5, or fearfully take a step back towards the couch with probability of 0.3. She will never go behind the couch. Therefore, if she is already at the couch, it has a 0.2 probability of stepping forward and a 0.8 probability of staying at the couch.

If you were to observe her for a very long time, for what proportion of the time will she be at the couch?